Main

Many fluids in nature and industry exhibit a nonlinear relationship between shear stress and shear rate, which is referred to as non-Newtonian behaviour. Several non-Newtonian features can exist, and they are often present simultaneously. In this Article we focus on the so-called elastoviscoplastic (EVP) fluids, which are fluids with elastic, viscous and plastic properties. EVP materials combine solid-like behaviour and fluid-like response depending on the value of the applied stress: they behave like a solid when the applied stress is below a critical value known as the ‘yield stress’, and flow like a liquid otherwise1. The elastic nature of these materials is present in their solid as well as liquid states2. Such fluids are common in everyday life (examples include toothpaste, jam, cosmetics and mud), and turbulent flows of EVP fluids are found in many industrial processes, including sewage treatment, crude oil transportation, concrete pumping and mud drilling3,4,5, as well as in nature as landslides and lava flows6,7.

A great deal of work has been done in the past to properly characterize the viscoelastic behaviour of a fluid in both laminar and turbulent flows8,9,10,11,12,13, while the effect of plasticity has been studied mainly in low-Reynolds-number laminar conditions1,14,15. Little is known about the plastic behaviour of an EVP fluid in turbulence. Rosti et al.16 studied a turbulent channel flow of an EVP fluid, finding that the shape of the mean velocity profile controls the regions where the fluid is unyielded, forming plugs around the channel centreline that grow in size as the yield stress increases, similar to what is observed in a laminar condition. However, the presence of the plug region has an opposite effect on drag for laminar and turbulent flow configurations, resulting in drag reduction in the turbulent case and drag increase in the laminar one; the turbulent drag behaviour is due to the tendency of the turbulent flow to relaminarize, overall leading to a strongly nonlinear relation between yield stress and drag coefficient. Simulation results were then employed by Le Clainche et al.17, using high-order dynamic mode decomposition, to study the near-wall dynamics, comparing them to those in Newtonian and viscoelastic fluids. Their work revealed that both elasticity and plasticity have similar effects on the near-wall coherent structures, where the flow is characterized by long streaks disturbed for short periods by localized perturbations. A recent experimental study by Mitishita et al.18 on a turbulent duct flow of Carbopol solution de facto verified the numerical results obtained by Rosti et al.16 on the effect of plasticity on the mean flow profile and Reynolds stresses. Additionally, they observed an increase in the energy content at large scales and a decrease at small scales, when compared with a Newtonian fluid. Mitishita et al. reported a −7/2 scaling in the energy spectra at high wavenumbers during Carbopol flows compared to −5/3 scaling in the case of water flows. The newly observed scaling was attributed either to the decrease in the inertial effect in the presence of Carbopol solutions, which shrinks the inertial range of scales because the Reynolds numbers are much lower than in water flows, or to the elastic effects that become important at large wavenumbers where the fluid experiences high frequencies. Moreover, the shear-thinning effects that Carbopol solutions exhibit affected the anisotropy and the overall flow behaviour. The elastic and shear-thinning effects are rheological features of Carbopol solutions and cannot be eliminated experimentally.

Homogeneous and isotropic turbulent flows have long been a focus of turbulence research for their simple theoretical analysis and the generality of their results. To this end, as has been extensively done in the past for viscoelastic flows, here we study tri-periodic homogeneous flow, where the celebrated K41 theory by Kolmogorov19 can be directly applied to a classical Newtonian fluid. In this Article we study a homogeneous isotropic turbulent (HIT) flow of an EVP fluid at high Reynolds number, as shown in Fig. 1. We aim to answer the following fundamental question: how does the Kolmogorov theory change when the fluid is EVP? We will mainly focus on its plastic behaviour and investigate how the yield stress affects the multiscale energy distribution and balance, and how the turbulent energy cascade is altered by the fluid’s plasticity. Our results show profound modifications of the classical picture predicted by the K41 theory for Newtonian fluids, with the emergence of a new scaling range, the dominance of the non-Newtonian flux and dissipation at small and intermediate scales, and enhanced intermittency of the flow.

Fig. 1: Instantaneous colourmaps of the turbulent fluid dissipation ϵf in homogeneous isotropic turbulence of an EVP fluid at different Bingham numbers.
figure 1

Yielded regions are shown with a black–red–yellow colourscale, and unyielded regions with black–grey–white.

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Results

To investigate the problem at hand, we performed massive three-dimensional direct numerical simulations (DNS) of HIT where we solve the flow equations fully coupled with the constitutive equation of the EVP fluid, within a tri-periodic domain of size L, using 1,024 grid points per side, as described in more detail in the Methods. The flow is controlled by four main parameters: the Reynolds number ReΛ, the Weissenberg number WiΛ, the viscosity ratio α and the Bingham number BiΛ, all based on the root-mean-square velocity fluctuations u′ and Taylor’s microscale Λ. We use the definitions \({{\rm{Re}}_{\varLambda }\equiv \rho {u}^{{\prime} }\varLambda /{\mu }_{{{{\rm{t}}}}},\,{\rm{Wi}}_{\varLambda }\equiv \lambda {\mu }_{{{{\rm{t}}}}}/\rho {\varLambda }_{0}^{2},\,\alpha ={\mu }_{{{{\rm{n}}}}}/{\mu }_{{{{\rm{t}}}}}}\) and \({\rm{Bi}}_{\varLambda }\equiv {\tau }_{\rm{y}}{\varLambda }_{0}/{\mu }_{{{{\rm{t}}}}}{u}_{0}^{{\prime} }\), where ρ is the fluid density, μt ≡ μf + μn is the total dynamic viscosity with μf being the fluid viscosity and μn the non-Newtonian one, λ is the relaxation time, τy is the yield stress, and subscript 0 denotes quantities from the BiΛ = 0 case. The Reynolds number describes the ratio of inertial to viscous forces, and we limit our analysis to high-Reynolds-number flows, achieving a Taylor microscale Reynolds number ReΛ ≈ 435 for the Newtonian flow, at which statistics of the flow have been found to be universal and exhibiting a proper scale separation, with an extensive inertial range of scales extended to almost two decades of wavenumbers. The Reynolds number explored here is the highest in DNS of HIT of non-Newtonian fluids. The Weissenberg number describes the ratio of elastic to viscous forces, and here we limit the analysis to WiΛ ≪ 1 (that is, WiΛ ≈ 10−3), to ensure that elastic effects are subdominant and all the observed changes are due to plasticity. We also fix a value of α = 0.1 to represent a dilute concentration of polymers, in accordance with previous works on the subject16,20. Thus, the key control parameter we vary is BiΛ, which describes the ratio of the yield stress to the viscous stress, and thus correlates with the prevalence of unyielded regions.

Figure 2 depicts the turbulent kinetic energy spectra of the cases analysed. The BiΛ = 0 case is similar to the Newtonian case shown in Supplementary Fig. 1, confirming that the effect of elasticity is subdominant and can be ignored. A clear E ~ κ−5/3 range is visible for more than one decade, showing that ReΛ is high enough to achieve scale separation, with the spectra exhibiting an inertial range of scales followed by a dissipative range. As BiΛ increases, the inertial range is limited to the large scales (small wavenumbers κ), with the energy increasing at large scales and decreasing at small scales. A clear deviation from Kolmogorov scaling becomes noticeable for BiΛ > 1, resulting in the emergence of a new apparent scaling of E ~ κ−2.3 that is shown more clearly by plotting compensated energy spectra (as shown in Supplementary Fig. 3). The difference in scaling between the experimental work (−7/2)18 and the current study (−2.3) is mainly due to the higher values of Reynolds and Bingham numbers considered here. The abrupt change in the spectra with BiΛ is consistent with the bulk flow properties (ReΛ and the volume fraction of the unyielded regions Φ) shown in the inset of Fig. 2: for the cases where BiΛ < 1, ReΛ remains relatively unaltered, with Φ always close to zero, whereas when BiΛ further increases, the microscale Reynolds number ReΛ and the volume Φ of the unyielded regions rapidly increase with a similar trend.

Fig. 2: Turbulent kinetic energy spectra of EVP flows with various Bingham numbers.
figure 2

Different BIngham numbers are plotted in colours from dark to light; BiΛ = 0, 0.0025, 0.025, 0.25, 2.5, 12.5 and 25 are plotted in black, purple, dark blue, light blue, dark green, light green and orange, respectively. The expected Kolmogorov scaling for a Newtonian fluid is shown by a grey dashed line, and the grey dash-dotted line represents an apparent new non-Newtonian scaling E ~ κ−2.3 that emerges at large BiΛ. Inset: variation of the mean values of the microscale Reynolds number ReΛ (right axis, plotted as squares) and the volume fraction of the unyielded regions Φ (left axis, plotted as diamonds) as a function of BiΛ. Error bars report the s.d. of ReΛ in time, measured using 103 samples. Plastic effects start to appear for BiΛ ≳ 1, suggesting that Λ is the relevant length scale of the problem.

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To fully characterize the change in the energy spectra, we study the turbulent kinetic energy balance, which in wavenumber space can be expressed as

$${{{{{\mathcal{F}}}}}_{{{{\rm{inj}}}}}(\kappa )+\varPi (\kappa )+{{{\mathcal{D}}}}(\kappa )+{{{\mathcal{N}}}}(\kappa )=\langle {\epsilon }_{{{{\rm{f}}}}}\rangle +\langle {\epsilon }_{{{{\rm{n}}}}}\rangle =\langle {\epsilon }_{{{{\rm{t}}}}}\rangle },$$
(1)

where \({{{{\mathcal{F}}}}}_{{{{\rm{inj}}}}}\) is the turbulence production introduced by the external forcing (injected at the largest scale, κL ≡ 2π/L), and \({\varPi ,\,{{{\mathcal{D}}}}}\) and \({{{\mathcal{N}}}}\) are the nonlinear energy flux, the fluid dissipation and the non-Newtonian contribution, respectively. In addition to the classical bulk fluid dissipation rate ϵf, here we have a non-Newtonian dissipation ϵn, which is the rate of removal of turbulent kinetic energy from the flow due to the non-Newtonian extra stress tensor (Supplementary Sections I and II provide a derivation of this equation). Figure 3 shows the turbulent kinetic energy balance for a few representative values of BiΛ. When comparing with Supplementary Fig. 1b, the BiΛ = 0 case closely follows the classical Newtonian turbulent flow, where energy is carried by Π from the large to small scales before being dissipated by the fluid viscosity \({{{\mathcal{D}}}}\). The contribution of the nonlinear convective term Π, which appears as an almost horizontal plateau at relatively large scales, progressively decreases with BiΛ and shrinks towards larger scales, consistent with the reduction of the extension of the inertial range observed in Fig. 2. The reduced energy flux with BiΛ is also accompanied by a decrease of the fluid dissipation \({{{\mathcal{D}}}}\), which is instead compensated by the increase of non-Newtonian contribution \({{{\mathcal{N}}}}\). At small scales (large κ), the relative importance of the non-Newtonian contribution increases with BiΛ, becoming comparable to the fluid dissipation for BiΛ ≈ 2.5 and eventually becoming the dominant term for BiΛ ≳ 12.5, corresponding to the emergence of the new scaling in the energy spectrum shown in Fig. 2; indeed, the non-Newtonian contribution can be interpreted as a combination of a pure energy flux (giving rise to the new scaling region) and a pure dissipative term, as recently suggested by Rosti and others21. Regarding the direction of energy flux, Supplementary Fig. 4 shows that we have a direct cascade of energy from large to small scales for all BiΛ (refs. 22,23).

Fig. 3: Scale-by-scale energy balance for different BiΛ.
figure 3

Plotted are the energy flux of the nonlinear convective term Π (dashed lines), solvent dissipation \({{{\mathcal{D}}}}\) (dotted lines) and the non-Newtonian contribution \({{{\mathcal{N}}}}\) (solid lines) for BiΛ = 0 (black), BiΛ = 2.5 (dark green), BiΛ = 12.5 (light green) and BiΛ = 25 (orange). Each term is normalized by the total dissipation rate 〈ϵt〉. \({{{\mathcal{N}}}}\) grows at intermediate and small scales when BiΛ is increased, eventually becoming the dominant contribution.

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We extend the analysis done in the spectral domain by computing the longitudinal structure functions defined as Sp(r) = 〈(Δu(r))p〉, where p is the order of the structure function and Δu(r) = u(x + r) − u(x) is the difference in the fluid velocity across a length scale r, projected in the direction of r. According to K41, \({S}_{{{{\rm{p}}}}}(r) \sim {\left(\langle {\epsilon }_{{{{\rm{t}}}}}\rangle r\right)}^{p/3}\); however, when the structure functions are displayed as a function of r, as shown in Fig. 4a, they deviate from the K41 prediction as p increases. This phenomenon is thought to be due to the intermittency of the flow, that is, extreme events that are localized in space and time, and thereby break Kolmogorov’s hypothesis of self-similarity in the inertial range24. Intermittency results in the scaling exponent of r being a nonlinear concave function of p (instead of p/3)25. For the EVP fluid, two scaling regions appear at large BiΛ, with scaling consistent with those from the energy spectra, and with intermittency present in both scaling regions. The role of intermittency in the scaling exponents can be better appreciated when the structure functions are displayed in their extended self-similarity form, obtained by plotting one structure function against another26. In Fig. 4b, S4 and S6 are plotted against S2 for all Bingham numbers considered. We note a clear power-law scaling, which deviates from Kolmogorov’s prediction, even for the BiΛ = 0 case (shown in black). The departure from Kolmogorov’s prediction progressively grows as the plasticity of the fluid increases, suggesting that the flow becomes more intermittent due to its plasticity. This becomes more obvious when we plot Sn compensated by the intermittency correction at BiΛ = 0 against S2 (Supplementary Fig. 5). Also, intermittency appears to act equally in the two scaling regions present at large BiΛ.

Fig. 4: Analysis of structure functions.
figure 4

a, Dependence of the longitudinal velocity structure functions S2 (pluses), S4 (squares) and S6 (hexagons) on separation distance r. Symbol colour denotes BiΛ and is the same as in Fig. 2. Dashed lines show scalings predicted by K41, and dash-dotted lines show scalings predicted using the new non-Newtonian scaling E ~ κ−2.3. b, The two scalings collapse onto a single line when we plot the structure functions S4 and S6 in extended self-similarity form, that is, against S2. To easily see changes in gradient, we have normalized the structure functions by their values at r ≈ Λ. The dotted line shows a best fit through the data for BiΛ = 0, which deviates from the K41 prediction (dashed line) due to intermittency. Increasing BiΛ further increases this deviation. The inset in b reports the multifractal spectrum of the energy dissipation rate carried out by the fluid ϵf.

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Intermittency originates from the multifractal nature of the turbulent dissipation rate24. For Newtonian fluids, this can be quantified by the multifractal spectrum of the energy dissipation rate, ϵf (refs. 24,27), which we report in the inset of Fig. 4b. This graph demonstrates that F(α) is nearly identical for all BiΛ cases except for minor variations at small and large values of α. This implies that the fluid dissipation rate is not the cause of the enhanced intermittency observed in the extended self-similarity analysis.

In the present flow, the turbulent kinetic energy is dissipated by two different terms, ϵf and ϵn, as seen in Fig. 1. We thus investigate their respective behaviour by looking at their probability distribution functions (PDFs; Fig. 5). We name the non-Newtonian contribution ϵn a ‘dissipation’ because, on average, it removes energy from the flow, giving rise to the positive-skewed distributions in Fig. 5b; however, unlike the fluid dissipation, it can take positive or negative values at particular locations in space and time. Figure 5a shows that the distribution of ϵf narrows as BiΛ increases28; on the other hand, from Fig. 5b, we see that that the distribution of ϵn substantially broadens as BiΛ increases. Because the non-Newtonian dissipation becomes dominant for the largest BiΛ (as shown in Fig. 3), we can thus infer that the extreme values of ϵn are indeed the source of the enhanced intermittency observed from the structure function analysis in Fig. 4.

Fig. 5: PDFs of the fluid dissipation rate ϵf and non-Newtonian dissipation rate ϵn.
figure 5

a,b, PDF of the fluid dissipation rate ϵf (a) and of the non-Newtonian dissipation rate ϵn (b) averaged over time, and normalized by 〈ϵ0〉, the total dissipation of the BiΛ = 0 flow. As BiΛ increases, the PDF of ϵf narrows slightly, whereas the PDF of ϵn widens substantially.

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Discussion

By means of unprecedented high-Reynolds-number DNS of an EVP fluid, we have shown that plastic effects substantially alter the classical turbulence predicted by Kolmogorov theory for Newtonian fluids.

We have proved that the non-Newtonian contribution to the energy balance becomes dominant at intermediate and small scales for large Bingham numbers, inducing the emergence of a new intermediate scaling range in the energy spectra between the Kolmogorov inertial and dissipative ranges, where the energy spectrum decays with a −2.3 exponent. Interestingly, this exponent has been recently found for the turbulence of viscoelastic fluids at large Reynolds and Weissenberg numbers21,29, suggesting a possible similarity among plastic and elastic effects on the turbulent cascade. This similarity in the scaling behaviour of the two cases could be attributed to a similar interaction mechanism in the Navier–Stokes equation between the convective and extra stress terms. It is also worth noting that in the context of viscoelastic flows at high Weissenberg number, an exponent less than or equal to −3 has been widely reported in the past8; however, this is only found at relatively lower Reynolds number than investigated here or explored in recent experimental and numerical work21,29. The present work reports the −2.3 scaling in turbulent flows of highly plastic EVP fluids, and further studies on the size and distribution of the unyielded regions could shed more light on the origin of the newly found scaling.

We have also shown that the flow in the presence of plastic effects is more intermittent than in a Newtonian fluid, due to the combination of the classical intermittency originating from the multifractal nature of the turbulent dissipation rate, which remains substantially unaltered, and a new plastic contribution that instead grows with the Bingham number. A direct consequence of this result is that intermittency corrections for an EVP fluid are non-universal and dependent on the flow configuration, differently from viscoelastic flows. These results are relevant for several catastrophic natural flows with high plasticity, such as lava flows and landslides30. Our findings explain why such flows are usually found to be intermittent and frequently aggressive, resulting in more damage. The non-universality of the flow intermittency in EVP fluids reflects also in an increased difficulty in their modelling.

Methods

Governing equations

The flow under investigation is governed by a system of a scalar, a vector and a tensorial equation—the incompressibility constraint, the conservation of momentum, and the constitutive equation for the non-Newtonian extra stress tensor, respectively. The incompressibility constraint and the momentum conservation equations can be written as

$${\nabla \cdot {{{\bf{u}}}}}={0},$$
(2)
$${\rho \left(\frac{\partial {{{\bf{u}}}}}{\partial t}+{{{\bf{u}}}}\cdot \nabla {{{\bf{u}}}}\right)=\nabla p+{\mu }_{{{{\rm{f}}}}}{\nabla }^{2}{{{\bf{u}}}}+{{{{\bf{f}}}}}_{{{{\rm{inj}}}}}+{{{{\bf{f}}}}}_{{{{\rm{evp}}}}}},$$
(3)

where u is the fluid velocity, p is the pressure, ρ is the density and μf is the fluid dynamic viscosity. The term finj represents the external force used to sustain turbulence; here we consider the Arnold–Beltrami–Childress (ABC) flow with forcing

$$\begin{array}{l}{{{{{\bf{f}}}}}_{{{{\rm{inj}}}}}}={{{{\bf{i}}}}{\mu }_{{{{\rm{f}}}}}(A\sin z/L+C\cos y/L)+{{{\bf{j}}}}{\mu }_{{{{\rm{f}}}}}(B\sin y/L+A\cos z/L)}\\\qquad+{{{\bf{k}}}}{\mu }_{{{{\rm{f}}}}}(C\sin y/L+B\cos x/L),\end{array}$$
(4)

where i, j and k are the Cartesian unit vectors, A, B and C are real parameters, and the flow has periodicity L in x, y and z. In our simulations, we choose A = B = C and use an appropriate value of μf to give a microscale Reynolds number \({{{{{\rm{Re}}}}}_{\varLambda }\approx 435}\) for the Newtonian flow. The last term in equation (3) is defined as fevp ≡ ∇ ⋅ τ, where τ is the non-Newtonian extra stress tensor of the EVP fluid. We adopt the constitutive model proposed by Saramito31 to express the evolution of the extra stress tensor, which can be written as

$${\lambda }\mathop{\uptau}\limits^{\nabla}+\max \left(0,\,\frac{{\tau }_{\rm{d}}-{\tau }_{\rm{y}}}{{\tau }_{\rm{d}}}\right){{{\mathbf{\uptau }}}}={\mu }_{{{{\rm{n}}}}}\left(\nabla {{{\bf{u}}}}+{\left(\nabla {{{\bf{u}}}}\right)}^{T}\right)$$
(5)

where (\(\mathop\cdot\limits^{\nabla}\)) denotes the upper convected derivative, that is, \(\mathop{\uptau}\limits^{\nabla}={\frac{\partial {{{\mathbf{\uptau }}}}}{\partial t}+{{{\bf{u}}}}\cdot \nabla {{{\mathbf{\uptau }}}}-{{{\mathbf{\uptau }}}}\cdot \nabla {{{\bf{u}}}}-{(\nabla {{{\bf{u}}}})}^{T}\cdot {{{\mathbf{\uptau }}}}}\). μn is the non-Newtonian dynamic viscosity, τd is the magnitude of the deviatoric part of the stress tensor \({{{{\mathbf{\uptau }}}}}_{\rm{d}}\equiv {{{\mathbf{\uptau }}}}-{{{\rm{tr}}}}({{{\mathbf{\uptau }}}}){{{\bf{I}}}}/3\), and I is the identity tensor, that is, \({\tau }_{\rm{d}}={\sqrt{\frac{1}{2}({{{{\mathbf{\uptau }}}}}_{\rm{d}}:{{{{\mathbf{\uptau }}}}}_{\rm{d}})}}\). Before yielding, that is, τd ≤ τy, the model predicts only recoverable Kelvin–Voigt viscoelastic deformation; after yielding, that is, τd > τy, it predicts Oldroyd-B viscoelastic behaviour. This transition occurs in a continuous manner. There are other EVP models that take into account shear-thinning32 or thixotropic behaviour33; however, we chose the one described above for its simplicity and the least number of involved parameters. Also, this model proved able to capture the main flow characteristics in a turbulent channel flow16,18.

Numerical method

We use the in-house flow solver Fujin (https://groups.oist.jp/cffu/code) to solve the governing equations numerically on a staggered uniform Cartesian grid. Velocities are located on the cell faces, and pressure, stresses and the other material properties are located at the cell centres. The second-order central finite-difference scheme is used for spatial discretization except for the advection term that comes from the upper convective derivative in equation (5), where the fifth-order WENO (weighted essentially non-oscillatory) scheme is adopted34. The second-order Adams–Bashforth scheme coupled with a fractional step method35 is used for the time advancement of all terms except for the non-Newtonian extra stress tensor, which is advanced with the Crank–Nicolson scheme. To enforce a divergence-free velocity field, a fast Poisson solver based on the fast Fourier transform is used for the pressure. The domain decomposition library 2decomp (http://www.2decomp.org) and the MPI protocol are used to parallelize the solver. The evolution equation of the extra EVP stress is formulated and solved using the log-conformation method36 to ensure the positive-definiteness of the conformation tensor. The fluid domain is a periodic cubic box of length L discretized using 1,024 grid points per side, resulting in a large grid resolution sufficient to represent the fluid properties at all the scales of interest with \({\eta /\Delta x}={{{{\mathcal{O}}}}(1)}\), where η is the Kolmogorov length-scale, and Δx is the grid spacing.