3.1.1. The NS equations projected on a logarithmic lattice

We first illustrate the above idea using a simplified model of fluid turbulence obtained by projecting the NS equations onto a Fourier lattice with exponentially spaced modes. The Fourier modes are given by $k_n = k_0 \lambda ^n$ , where $\lambda$ is the log-lattice parameter (chosen to be $2$ or $\phi = {1+\sqrt 5}/2$ in our simulations). This projection restricts the number of possible interactions between modes, but obeys the symmetries and the global conservation laws associated with the fluid flow. For detailed theoretical and numerical discussions we refer the reader to Campolina & Mailybaev (Reference Campolina and Mailybaev2018, Reference Campolina and Mailybaev2021), Campolina (Reference Campolina2019) and Pikeroen et al. (Reference Pikeroen, Barral, Costa, Campolina, Mailybaev and Dubrulle2024).

Figure 1. Scaling laws of turbulence on log-lattice. (a) The RMS velocity as a function of Reynolds number. The data are the symbols. The black dashed line is the fit, assuming a power law for the efficiency, with corresponding $R^2 = 0.9987$ . (b) Dimensionless dissipation as a function of Reynolds number. The data are the symbols. The black dotted line is the theoretical bound $\sqrt {2}/{\mathcal{E}}$ , and the dashed line is the fit $\epsilon _*/{\mathcal{E}}^{3/2}$ , with corresponding $R^2 = 0.9960$ . (c) Viscosity (inverse Reynolds number) versus inefficiency (inverse efficiency). A second-order phase transition is observed where viscosity vanishes at finite inefficiency (Costa et al. Reference Costa, Barral and Dubrulle2023). Colour codes the forcing type: orange, constant power; blue, constant forcing amplitude. The black dashed line is a fit, assuming a power law for the efficiency, with $R^2=0.9576$ . These log-lattice simulations were done with $\lambda =2$ and the insets with $\lambda =\phi$ .

In figure 1 we show our results obtained from the log-lattice simulations of (2.1) in three dimensions with a forcing $\boldsymbol{f}$ of amplitude $F_0$ around a wavenumber $k_f=2\pi /L_f$ for different values of viscosity. In these simulations, we use a dyadic spacing $\lambda =2$ and a golden number spacing $\lambda =\phi =1.618$ with a maximal wavenumber $k_{{max}}$ up to $3 \times 10^6$ . We accumulate various statistics of Fourier modes once a statistically stationary state has been achieved. In particular, we define the kinetic energy $E$ and the energy injection as

(3.2) \begin{equation} E=\frac {1}{2}\sum _k \overline {u(k,t)u(-k,t)} \quad \text{and} \quad \epsilon =\sum _k \overline {u(k,t)f(-k,t)+u(-k,t)f(k,t)},\\ \end{equation}

respectively, where the overline denotes a time average. Note that in a stationary state, $\epsilon$ also represents the mean energy dissipation and $u_{\textit{rms}}=\sqrt {2E}$ represents the root-mean-square (RMS) velocity, since the mean flow is zero. We use these quantities to define the Reynolds number ${\textit{Re}}=L_f\sqrt {2E}/\nu$ , the dimensionless energy injection/dissipation $D_{\epsilon }=\epsilon L_f / E^{3/2}$ and the efficiency ${\mathcal{E}}=E L_f / F_0$ (also see table 1).

Figure 1(a) shows that despite attaining fairly high ${\textit{Re}}$ in our log-lattice simulations, the RMS velocity is yet to reach a saturation, believed to be a hallmark of fully developed turbulence. Similarly, the dimensionless energy dissipation $D_{\epsilon }$ continues to decrease as ${\textit{Re}}$ is increased; however, near the largest ${\textit{Re}}$ the decay rate has decreased, but $D_{\epsilon }$ is still far from saturating. Note that the saturation of the energy dissipation to a constant value is a hallmark of the dissipation anomaly. We also observe that for the ${\textit{Re}}$ considered here, the bound $D_\epsilon \leqslant \sqrt {2}/{\mathcal{E}}$ given by (2.8) is always satisfied. In figure 1(c) we show the behaviour of $1/{\textit{Re}}$ versus $1/{\mathcal{E}}$ for the dyadic lattice. The saturation curve can be very well fitted by an implicit power law as

(3.3) \begin{equation} \frac {1}{{\textit{Re}}}=A \big({\mathcal{E}}^{-1}-{{\mathcal{E}}_*}^{-1} \big)^\gamma , \end{equation}

with $A=10^{-4.86}$ , ${\mathcal{E}_*}=55$ and $\gamma =3.45$ . The efficiency saturation in turn provides a dissipation anomaly, following the law with leading-order asymptotic expansion (see figure 1 b):

(3.4) \begin{equation} D_\epsilon = \frac {\epsilon _*}{{\mathcal{E}}^{3/2}} = \frac {\epsilon _*}{{\mathcal{E}}_*^{3/2}}\left (1+\frac 32\frac {{\mathcal{E}}_*}{(A{\textit{Re}})^{1/\gamma }}\right )+o({\textit{Re}}^{-1/\gamma }). \end{equation}

Power laws of such type are found in phase transition or in bifurcation, and it is not accidental that we find it here. It was indeed shown in Costa et al. (Reference Costa, Barral and Dubrulle2023) that solutions of the RNS equations projected on log-lattices display a bifurcation at the value $\mathcal{E}_*$ : above $\mathcal{E}_*$ , only self-similar blow-up solutions of Euler equations are observed. Below $\mathcal{E}_*$ , dissipative irreversible solutions of the NS equations are observed. Such bifurcation can be observed in the behaviour or the enstrophy, that diverges above $\mathcal{E}_*$ , while staying finite below $\mathcal{E}_*$ . Considering the equivalence of NS and RNS, we thus conclude that the power law we find in our NS case is just the counterpart of the power law observed in RNS. Note, however, that in the NS case, we do not have a phase transition, since we observe only one ‘phase’, the dissipative irreversible solutions of NS equations. The only caveat is that the warm phase where thermalisation occurs is also accessible for NS by using an under-resolved Galerkin truncated direct numerical simulation (DNS).

The case $\lambda =\phi$ can also be well described by a power law. Specifically, we obtain a good fit for the efficiency using the following set of coefficients: $A=10^{-0.85}$ , ${\mathcal{ E}_*}=1.27$ and $\gamma =3.45$ (see the inset of figure 1 c). This may only be two cases, but the different forcing and lattice spacing suggest that the scaling exponents are universal.

Now, once we obtain the fit, we can invert (3.3) to get $u_{\textit{rms}}$ as a function of ${\textit{Re}}$ . We show this in figure 1(a) using black dashed lines. The extrapolation to large Reynolds numbers indicates that the saturation is not observed until ${\textit{Re}}\sim 10^{12}$ , which is considerably beyond our computational capacity. Similarly, we can use this law to fit the variation of the energy dissipation with ${\textit{Re}}$ in figure 1(b) and extrapolate predictions for small viscosities. We find that a good fit to the data is given by $D_\epsilon =\epsilon _*/{\mathcal{E}}^{3/2}$ , where ${\mathcal{E}}({\textit{Re}})$ is obtained by inverting (3.3) with constant $\epsilon _*=53$ ( $\epsilon _*=8$ ) for $\lambda =2$ ( $\lambda =\phi$ ). The extrapolated black dashed curve suggests that the dimensionless energy dissipation becomes constant in the limit of vanishing viscosity, which is consistent with the existence of dissipation anomaly (the so-called zeroth law of turbulence).

3.1.2. Direct numerical simulation of RNS equations

We now repeat the analysis for the homogeneous isotropic turbulence obtained using the DNS of the RNS equations over a triply period domain of size $(2\pi )^3$ . For more details we refer the reader to Shukla et al. (Reference Shukla, Dubrulle, Nazarenko, Krstulovic and Thalabard2019). In RNS, the imposition of a global constraint of the conservation of a quadratic quantity, for example the total energy, allows for the viscosity $\nu (t)$ to fluctuate in time; however, its statistical properties are equivalent to those obtained using the standard INSE (2.1). We use two different codes: S1, a classical pseudo-spectral code with $64^3$ and $128^3$ collocation points, and S2, a pseudo-spectral code with Taylor–Green symmetries that allows us to perform DNS with $512^3$ and $1024^3$ collocation points. Note that the Reynolds numbers reached here are considerably smaller than those of log-lattice simulations.

Figures 2(a) and 2(b) show that despite the comparatively smaller Reynolds numbers, both $u_{\textit{rms}}/U_0$ and the dimensionless dissipation, respectively, begin to saturate for ${\textit{Re}} \simeq 10^4$ . Similarly $1/{\textit{Re}}$ starts to show saturation starting at ${\mathcal{E}}_* \sim 0.2$ (see figure 2 c). A fit of the form (3.3) is obeyed with the exponent $\gamma =1.4$ and the prefactor $A=10^{-0.2}$ . Inversion of this power law allows us to obtain a parameter-free fit of the kinetic energy in figure 2(a) and a one-parameter fit of the energy dissipation $2.5{\mathcal{E}}^{-3/2}$ . In summary, the DNS results are very similar to the case of the flows projected on log-lattices, but belong to a different ‘universality class’ (laws have the same shape but different constants). Given the equivalence between RNS and NS, we can trace this difference to the number of allowed triadic interactions in Fourier space.

Figure 2. Scaling laws of turbulence in numerical homogeneous isotropic turbulence. (a) The RMS velocity as a function of Reynolds number. To account for different forcing type, we have divided the RMS velocity by $U_0=\sqrt {F_0 L_f}$ . The data are the symbols. The black dashed line is the fit, assuming a power law for the efficiency, with $R^2 = 0.9844$ . (b) Dimensionless dissipation as a function of Reynolds number. The data are the symbols. The dashed line is the fit $2.5/{\mathcal{E}}^{3/2}$ , with $R^2= 0.8733$ . (c) Viscosity (inverse Reynolds number) versus inefficiency (inverse efficiency). Red and yellow symbols: S1 at resolution $64^3$ and $128^3$ ; magenta and blue symbols: S2 at resolution $512^3$ and $1024^3$ . The black dashed line is a fit, assuming a power law for the efficiency, with $R^2=0.9706$ . Here ${\textit{Re}}$ is not explored as far as in log-lattice simulations, but the phase transition/bifurcation begins at smaller ${\textit{Re}}$ , with a larger inefficiency compared with log-lattice simulations.

3.1.3. Comment on the dissipation scaling

Through the efficiency lens, we are able to connect energy dissipation and the Reynolds number by inverting the power law, resulting in (3.4). It is instructive to compare the asymptotic behaviour of the dissipation anomaly with other simple models. A qualitatively similar formula can be obtained by a mean field approach:

(3.5) \begin{equation} D_\epsilon = D_{\epsilon ,\infty }\left (\sqrt {1+\frac {3b^3}8\frac 1{{\textit{Re}}^2}}+\sqrt {\frac {3b^3}8}\frac 1{{\textit{Re}}}\right ) = D_{\epsilon ,\infty }\left (1+\sqrt {\frac {3b^3}8}\frac 1{{\textit{Re}}}\right )+o({\textit{Re}}^{-1}), \end{equation}

where $b$ is the Kolmogorov constant for the two-point structure constant. A full derivation can be found in Lohse (Reference Lohse1994). This theoretical approach has the benefit of predicting that $\gamma =1$ . This is not specific to the chosen closure model of the article akin to a turbulent viscosity:

(3.6) \begin{equation} \nu _t = \frac {D^2}{b^3\epsilon }, \end{equation}

where $D$ is the two-point structure function. Any closure at the level of the two-point correlation function will necessarily give $\gamma =1$ . This follows from the Kármán–Howarth equation itself. In the inviscid case, the third-order structure function expressed in terms of $D,\epsilon$ but independent of $\nu$ (given by the choice of closure) balances the dissipation term, with a finite dissipation rate solution. The viscous term is proportional to $\nu$ , and will therefore generically give a $O(\nu )=O({\textit{Re}}^{-1})$ correction to the asymptotic dissipation rate. In statistical mechanics, the analogue would be the Landau theory whose structural limitations only allow it to predict mean-field critical exponents. To reproduce the numerical data, where $\gamma =3.4$ for log-lattice or $\gamma =1.4$ for DNS, the full hierarchy of structure functions seems to be necessary, requiring at least some analogue of the renormalisation group to recover the anomalous scaling.

3.2.1. Von Kármán flows

First, we consider the von Kármán flow generated between two counter- or co-rotating impellers in a cylindrical tank. The experimental set-up has impellers of radius $R_d$ in a tank of radius $R$ filled with a fluid of density $\rho$ and viscosity $\nu$ . The distance between the two impellers is $H=1.8 R$ . The fluid is forced and maintained in a turbulent state at a high Reynolds number by means of the two independently rotating impellers at the same constant frequency $F$ , located at the top and the bottom of the cylindrical tank. Flows with different Reynolds numbers are achieved by using different fluids: pure glycerol, mixture of glycerol and water, and water at $T=20\,^{\circ}{\textrm C}$ . In all the cases, the temperature is kept constant through a thermal regulation, to avoid drifts of viscosity during operation. The mean kinetic energy of the flow can be computed on a meridian plane, including the rotation plane, with the help of the stereoscopic particle image velocimetry technique, which allows for the reconstruction of the three components of the velocity within this plane. For further technical details about these measurements, we refer the reader to Monchaux (Reference Monchaux2007), Debue (Reference Debue2019) and Cortet et al. (Reference Cortet, Diribarne, Monchaux, Chiffaudel, Daviaud and Dubrulle2009, Reference Cortet, Chiffaudel, Daviaud and Dubrulle2010).

The axisymmetry of the flow ensures that the planar estimate is equal to the volume average of the kinetic energy. Moreover, inserting two torque meters, before and below the top and bottom impellers, respectively, allows for the measurement of the torque $\varGamma (t)$ applied by the turbulent flow on the impeller; this provides an estimate of the amplitude of the forcing per unit mass required to maintain the flow in a stationary state. The energy dissipation rate is measured using the mean injection rate, obtained by computing the mean work done by the torque $W=\overline {\varGamma (t) F}$ , where the overline denotes a time average. The choice of the unit of length $L=R$ and the unit of velocity $U=2\pi RF$ allows us to combine the parameters summarised in table 2 into the mean torque $K_p$ , the kinetic energy $E$ and the global Reynolds number ${\textit{Re}}_F$ in dimensionless form as follows:

(3.7) \begin{equation} K_p = \frac {\overline {\varGamma }}{ \rho \pi R^5 (2\pi F)^2}, \quad E = \frac {\langle \boldsymbol{u}\boldsymbol{\cdot }\boldsymbol{u}\rangle ^{1/2}}{2 (2\pi R F)^2} \quad \text{and} \quad {\textit{Re}}_F = \frac {2\pi R^2 F}{\nu }, \end{equation}

respectively, where $\langle \boldsymbol{\cdot }\rangle$ denotes an average over the meridian plane. In that experimental configuration, the mean torque provides two points of information. First, it directly provides the mean force applied to the flow by the propellers, providing an estimate of $F_0$ . Second, it can also provide the energy injection/dissipation $\epsilon$ in experimental conditions where the velocities of the upper and lower impellers are maintained constant in time. Indeed, in such a case, the mean power injected at the top and the bottom is computed as the time average of the torque times the frequency of the impeller. The latter being constant, the power is just the product of the mean torque times the (constant) frequency. We are thus in a situation where the equality in (2.8) is realised, so that the two quantities tend to zero or saturate together.

Table 2. Parameters of the von Kármán experiment.

Figure 3. Measurements in von Kármán flow with counter-rotating impellers. (a) Kinetic energy $E$ versus global Reynolds number ${\textit{Re}}_F$ . Grey symbols correspond to the real data from particle image velocimetry, while the blue symbols interpolate the missing data using a supercritical law $E \sim ({\textit{Re}}_F-3500)^{1/2}$ (Ravelet, Chiffaudel & Daviaud Reference Ravelet, Chiffaudel and Daviaud2008). (b) Mean dimensionless torque $K_p$ versus ${\textit{Re}}_F$ as measured by Ravelet (Reference Ravelet2005) (grey circles). The red triangles correspond to points measured in the SHREK experiments (superfluid helium) using eight-bladed impellers with curvature $72^\circ$ . (c) Efficiency of impellers of various radii, fitted with blades of curvature angle $\alpha$ .

Figure 4. Scaling laws of turbulence in experimental von Kármán flow with counter-rotating TM60+ impellers. (a) The RMS velocity as a function of Reynolds number. To account for different forcing amplitude, we have divided the RMS velocity by $U_0=\sqrt {F_0 L_f}$ . The data are the blue symbols. The black dashed line is the fit, assuming a power law for the efficiency, with $R^2= 0.9915$ . (b) Viscosity (inverse Reynolds number) versus inefficiency (inverse efficiency). The red triangles correspond to points measured in the SHREK experiments (superfluid helium) using eight-blade impellers (Saint-Michel et al. Reference Saint-Michel2014). The black dashed line is the fit, assuming a power law for the efficiency, with $R^2= 0.9730$ .

We analyse the data from the experimental runs, wherein the counter-rotating impellers are fitted with $16$ curved blades of angle $72^\circ$ in the scooping direction, resulting in a symmetric large-scale circulation (Ravelet Reference Ravelet2005; Cortet et al. Reference Cortet, Chiffaudel, Daviaud and Dubrulle2010); such impellers are referred to as TM60+. In figures 3(a), 3(b) and 3(c) we show plots of $E$ , $K_p$ and $\mathcal{E}_*$ , respectively, using the available data. We have sparse data points for the kinetic energy in the Reynolds number range ${\textit{Re}}_F \in [10^2, 10^6]$ and a detailed set of measurements of the dimensionless torque $K_p$ for the range ${\textit{Re}}_F \in [10^2, 10^5]$ . To be able to combine both the measurements, we first interpolate the kinetic energy over the range of Reynolds number for which the detailed measurements of $K_p$ exist. We perform the interpolation using the information that the transition from the laminar to turbulent state is supercritical, with an exponent 2 and critical Reynolds number ${\textit{Re}}_*=3500$ (see Ravelet et al. Reference Ravelet, Chiffaudel and Daviaud2008). Following this, we compute all the quantities of interest (in dimensionless units):

(3.8) \begin{equation} F_0 = \frac {K_p R^2 L_f }{ R_d^2}, \quad u_{\textit{rms}} = \sqrt {2E} \quad \text{and} \quad R_{\textit{rms}} = u_{\textit{rms}}L_f {\textit{Re}}_F, \end{equation}

where $L_f$ is the dimensionless forcing scale. For the rest of this discussion, we take $L_f$ to be unity. We show the results of the analysis similar to the numerical data in figure 4. We do not observe any clear saturation of the quantities $u_{\textit{rms}}/U_0$ , $D_{\epsilon }=1/{\mathcal{E}}$ and $1/{\textit{Re}}$ . However, if we assume a power law for the efficiency (3.3), we find a good fit for the parameters $A=10^{-2.7}$ , ${\mathcal{E}}_*=0.97$ and $\gamma =1.4$ , which is similar to the homogeneous isotropic turbulence case. Furthermore, we can use this fit to extrapolate the behaviour at higher Reynolds numbers. However, the above discussed experimental runs do not have torque data at further higher Reynolds numbers to check the quality of the extrapolation for the efficiency. Therefore, we use the torque data at very large Reynolds numbers measured in the SHREK facility (Rousset Reference Rousset2014; Saint-Michel Reference Saint-Michel2014; Diribarne Reference Diribarne2021). Note that the SHREK set-up has the same geometry as the experimental set-up discussed above, except that the impellers are fitted with $8$ blades instead of $16$ . We show those data in figure 4 using red filled triangles, which lie on the extrapolated behaviour (black dashed line).

We check the dependence of $\mathcal{E}_*$ with respect to the forcing details. To achieve this, we compute the efficiency at ${\textit{Re}}_F \sim 10^6$ based on the measurements of $K_p$ and the kinetic energy as reported in Ravelet (Reference Ravelet2005), wherein these quantities were obtained for impellers of various radii and fitted with blades of different curvature angle $\alpha$ . We find that the data are somewhat scattered, but on an average lie around the value ${\mathcal{E}_*} \sim 1$ , which shows that this value is independent of this geometry. This is rather remarkable, since the value of $\alpha$ has a profound impact on the flow anisotropy, as measured by the ratio between the poloidal and the toroidal energy (Ravelet Reference Ravelet2005).

3.2.2. Pipe flows

The second experimental flow that we consider is an example of a wall-bounded parallel flow. This is characterised by a mean flow that depends only on one space coordinate; as a consequence, streamlines are parallel. We choose a circular pipe flow and analyse the data available from the experimental works of Swanson et al. (Reference Swanson, Julian, Ihas and Donnelly2002) and McKeon et al. (Reference McKeon, Li, Jiang, Morrison and Smits2004). The flow is driven by a pressure gradient along the axis of the pipe, $\partial _x P$ , and results in a mean velocity $\overline {U}$ . In smooth pipes, pressure gradients balance the shear stress, so that the force applied to the flow can be measured either through the pressure drop along the pipe or through the friction drag $\tau _{w}$ . For the given fluid density $\rho$ , the viscosity $\nu$ and the pipe diameter $D$ , we can build two dimensionless parameters:

(3.9) \begin{equation} \lambda = 8\frac {-(\partial _x P)D}{ \rho \overline {U}^2} \quad \text{and} \quad {\textit{Re}} = \frac {\overline {U} D}{\nu }. \end{equation}

We show plots of $\lambda$ versus ${\textit{Re}}$ and $1/{\textit{Re}}$ versus $\mathcal{E}$ in figures 5(a) and 5(b), respectively. The force applied to the flow is $F_0=(1/\rho ) \partial _x P$ . We further assume that the typical forcing wavenumber is $k_f=1/D$ . To compute the efficiency, we need the kinetic energy $E$ stored in the flow. It can be split into two contributions, following $E=\overline {U}^2/2+ u^2_{\textit{rms}}/2$ . The first term is due to the presence of a mean-flow velocity and the second term is associated with the RMS velocity $u_{\textit{rms}}$ . However, we have $u^2_{\textit{rms}} \ll \overline {U}^2$ because of the presence of a large mean flow. As a consequence we can approximate the kinetic energy as $E=\overline {U}^2/2$ . Therefore, with this convention, we have $\lambda \sim 1/{\mathcal{E}}$ . In figure 5(b) we show the behaviour of $1/{\textit{Re}}$ versus $\lambda$ (or $\mathcal{E}$ ), where we observe that it follows the power law (3.3) with an exponent $\alpha =3.5$ . Clearly, this exponent is larger than the one that we obtained for homogeneous isotropic turbulence, which shows a sensibility to anisotropy and inhomogeneity.

When we invert the efficiency power law, we observe a saturation in the behaviour of $\lambda$ (see figure 5 a). This is in stark contrast to the Prandtl formula which predicts a decaying behaviour:

(3.10) \begin{equation} \lambda =\left [0.8382\;W(0.6287\; {\textit{Re}})\right ]^{-2} \sim \frac 1{(0.8382\ln {\textit{Re}})^2}, \end{equation}

where $W$ is the Lambert function (Dubrulle Reference Dubrulle2024). The fit displayed in figure 5(a) shows that effect of saturation will only be felt at ${\textit{Re}}\sim 10^{12}$ , a value way too large for present experimental facilities.

Figure 5. (a) Friction factor as a function of ${\textit{Re}}$ in pipe flow. The black dotted line is obtained by inverting the power law for efficiency, with $R^2= 0.9936$ . The blue dotted line represents the Prandtl-type formula with $R^2= 0.9966$ . The quality of the fit shows that for these data both laws have similar accuracy. Only experimental data at much higher Reynolds number could discriminate between the two. (b) Check of the power law for the efficiency. The dotted line is a fit with a power law of slope $3.5$ and $R^2=0.9966$ .

3.2.3. Taylor–Couette flow

Taylor–Couette flow is obtained by the co- or counter-rotation of two independent coaxial cylinders, with radii $a$ for the inner cylinder and $b$ for the outer cylinder. Many different experiments have been built to study such flow (Paoletti et al. Reference Paoletti, van Gils, Dubrulle, Sun, Lohse and Lathrop2012). We use here data from the Twente turbulent TC facility T $^3$ C, kindly provided to us by D. Lohse. The experiment is described in great detail in van Gils et al. (Reference van Gils, Bruggert, Lathrop, Sun and Lohse2011). It is characterised by cylinders of height $L = 92.7$ cm, and respective radii $a = 20.0$ cm and $b = 27.94$ cm. The maximum inner and outer angular velocities are $\varOmega _1/2\pi = 20$ Hz and $\varOmega _2/2\pi = \pm 10$ Hz, respectively. The torque is measured over the middle part of the inner cylinder of height $L_{\textit{mid}} = 53.6$ cm to minimise the influence of the end-plates (similar to Lathrop et al. Reference Lathrop, Fineberg and Swinney1992a ,Reference Lathrop, Fineberg and Swinney b ). This allows for the definition of the dimensionless torque $G = T/\rho \nu ^2 L_{\textit{mid}}$ , where $T$ is the torque, $\rho$ the fluid density and $\nu$ the kinematic viscosity.

The Taylor–Couette flow is characterised by three essential parameters (Dubrulle et al. Reference Dubrulle, Dauchot, Daviaud, Longaretti, Richard and Zahn2005), namely:

1. (i) its Reynolds number ${\textit{Re}}= ({2}/{1+\eta })\vert \eta {\textit{Re}}_2 - {\textit{Re}}_1\vert$ ;

2. (ii) its rotation number $R_\varOmega = (1-\eta )({{\textit{Re}}_1 + {\textit{Re}}_2}/{\eta {\textit{Re}}_2 - {\textit{Re}}_1})$ ;

3. (iii) its curvature number $R_{\mathcal{C}}= ({1-\eta }/{\eta ^{1/2}})$ ;

where ${\textit{Re}}_1=a\varOmega _1d/\nu$ and ${\textit{Re}}_2=b\varOmega _2d/\nu$ are the Reynolds number of the inner and outer cylinder. The above control parameters have been introduced so that their definitions apply to rotating shear flows in general and not only to the Taylor–Couette geometry. It is very easy in this formulation to relate the Taylor–Couette flow to shearing sheet (plane Couette flow with rotation) by simply considering the limit $R_{\mathcal{C}}\rightarrow 0$ .

Torque measurements provide the force intensity applying to the turbulent flow. Moreover, like in the von Kármán case, it also provides the power through the time average of the product of torque times the frequency. Since frequencies of the inner and outer cylinders are time-independent, we are in a situation where efficiency and energy injection follow the equality in (2.8).

Applying the definition (2.6), we thus obtain that in the case of Taylor–Couette flow, the efficiency and the dimensionless dissipation are given by ${\mathcal{E}}=({1}/{D_\epsilon })=({{\textit{Re}}^2}/{G})$ . Saturation of the efficiency or of the dimensionless dissipation corresponds to a situation where $G\propto {\textit{Re}}^2$ . This is called the ultimate regime, in which transport properties no longer depend on the molecular viscosity. Detailed analyses of $G$ as a function of ${\textit{Re}}$ and $R_\varOmega$ have been performed in Dubrulle et al. (Reference Dubrulle, Dauchot, Daviaud, Longaretti, Richard and Zahn2005) and Paoletti et al. (Reference Paoletti, van Gils, Dubrulle, Sun, Lohse and Lathrop2012). It was found that for all existing experimental data, it was possible to factorise the torque as $G=f(R_\varOmega )G_i$ , where $f$ is a function that only depends on $R_\omega$ and $G_i$ is the torque computed in a situation where the outer cylinder is at rest. Using this property, we can also factorise the efficiency and the dimensionless energy injection (friction factor) as

(3.11) \begin{equation} {\mathcal{E}}=\frac {1}{D_\epsilon }=\frac {1}{f(R_\varOmega )}\frac {{\textit{Re}}^2}{G_i}. \end{equation}

This shows that it is sufficient to consider only the behaviour of ${\textit{Re}}^2/G_i$ as a function of ${\textit{Re}}$ . This behaviour is reported in figure 6, for a case where only the inner cylinder is rotating. One sees that the friction factor does not saturate, but steadily decays with increasing ${\textit{Re}}$ , in the range of ${\textit{Re}}$ considered. Even though the range of Reynolds number is not very large, we have tried to see whether the reduced efficiency could be fitted by a power law (3.3) with $A=10^{-0.73466}$ , ${\mathcal{E}_*}=44$ and $\gamma =1.634$ . The result is shown in figure 6(b). The fit is clearly not as good as for previously considered cases, and it may well be that the efficiency never saturates in this case. Nonetheless, more data would be needed to draw a conclusion.

Figure 6. Reduced efficiency and friction factor in Taylor–Couette flow, obtained in a situation where only the inner cylinder is rotating. (a) Friction factor as a function of ${\textit{Re}}$ in Taylor–Couette flow. (b) Check of the power law for the efficiency. The dotted line is a power law with exponent $1.634$ . The fit was performed using only fully turbulent cases, i.e. ${\textit{Re}}\gt 2\times 10^3$ , giving $R^2=0.9892$ . Data are from the Twente turbulent TC facility T $^3$ (van Gils et al. Reference van Gils, Bruggert, Lathrop, Sun and Lohse2011). Data courtesy of D. Lohse.

3.2.4. Rayleigh–Bénard convection

The Rayleigh–Bénard setting corresponds to a situation where the fluid is heated from below, and enclosed between two plates, separated by a distance $H$ . The temperature difference between the two plates is $\Delta T$ . This classical setting has been the topic of many numerical and experimental studies (Roche Reference Roche2020; Lohse & Shishkina Reference Lohse and Shishkina2024). This is a paradigmatic example of a flow forced by buoyancy. The Rayleigh–Bénard flow is characterised by three dimensionless control parameters, namely:

1. (i) its Rayleigh number $\textit{Ra}= ({g\beta \Delta T H^3}/{\kappa \nu })$ ;

2. (ii) its Prandtl number $Pr={\nu }/{\kappa }$ ;

3. (iii) its aspect ratio $\varGamma ={H}/{L}$ ;

and two dimensionless response parameters:

1. (i) its Reynolds number ${\textit{Re}}= {UH}/{\nu }$ ;

2. (ii) its Nusselt number (dimensionless heat flux $\textit{Nu}=J H/\kappa \Delta T$ );

where $J$ is the heat flux, $U=\sqrt {2E}$ the typical velocity, $g$ the gravitational acceleration, $\beta$ the coefficient of thermal expansion, $\nu$ the kinematic viscosity, $\kappa$ the thermal diffusivity and $L$ a typical horizontal scale. In the case of no-slip boundary conditions, the variation of $\textit{Nu}$ and ${\textit{Re}}$ can been approximated with effective scaling laws, $\textit{Nu}\sim \textit{Pr}^\alpha \textit{Ra}^{\gamma }\!$ , for which the prefactor and effective exponent depend on the range in $\textit{Ra}{-}\textit{Pr}{-}\varGamma$ parameter space. All experiments and numerical simulations reported so far show $\gamma \lt 1/2$ (Lohse & Shishkina Reference Lohse and Shishkina2024). Through a recent theoretical and experimental effort, the aspect ratio dependence of the heat transfer in a cylindrical volume of various aspect ratios has been elucidated in Ahlers et al. (Reference Ahlers2022), and shown to be amenable to the introduction of a new effective height $H_{\textit{eff}}=H(\varGamma ^2/(1/49+\varGamma ^2)^{1/2}$ . An example of data collapse of different aspect ratios and $\textit{Pr}$ is shown in figure 7(a) for numerical (Scheel & Schumacher Reference Scheel and Schumacher2017; Stevens et al. Reference Stevens, Blass, Zhu, Verzicco and Lohse2018) and experimental (Chavanne et al. Reference Chavanne, Chillà, Castaing, Hébral, Chabaud and Chaussy1997; Methivier et al. Reference Méthivier, Braun, Chillà and Salort2022) data at various $\varGamma$ and $\textit{Pr}$ . There is a mild decay of $\textit{Nu}/\textit{Ra}^{1/3}$ , followed by a steeper rise at larger Rayleigh numbers, evidencing the transition between $\gamma \lt \sim 1/3$ and $\gamma \gt 1/3$ as the Rayleigh number increases. These data have been selected because they offer measurements of both $\textit{Nu}$ and ${\textit{Re}}$ as a function of $\varGamma$ , $\textit{Pr}$ and $\textit{Ra}$ , a feature that will be important in the computation of the efficiency (see below). However, they are typical of all data gathered so far. See Ahlers et al. (Reference Ahlers2022) for example of collapse with more data.

Figure 7. Scaling laws of turbulence in experimental and numerical Rayleigh–Bénard convection. To account for different aspect ratio, we have used everywhere the effective length scale $H_{\textit{eff}}=H(\varGamma ^2/(1/49+\varGamma ^2)^{1/2}$ (Ahlers et al. Reference Ahlers2022). Filled symbols: experiments by Chavanne et al. (Reference Chavanne, Chillà, Castaing, Hébral, Chabaud and Chaussy1997) and Methivier et al. (Reference Méthivier, Braun, Chillà and Salort2022); open squares: numerical data by Stevens et al. (Reference Stevens, Blass, Zhu, Verzicco and Lohse2018); open circles: numerical data by Scheel & Schumacher (Reference Scheel and Schumacher2017). The data are coloured according to $\log _{10}(\textit{Pr})$ . (a) Rescaled dimensionless heat transfer as a function of Rayleigh number. (b) Dimensionless kinetic energy dissipation as a function of Reynolds number. (c) Dimensionless thermal energy dissipation as a function of Péclet number. Data courtesy of O. Shishkina.

Theories explaining scaling laws in Rayleigh–Bénard flow are generally based on two exact relations for the kinetic and thermal energy dissipation (Shraiman & Siggia Reference Shraiman and Siggia1990; Lohse & Shishkina Reference Lohse and Shishkina2024):

(3.12) \begin{eqnarray} \epsilon _k =\nu ^3(\textit{Nu}-1) \textit{Ra} \textit{Pr}^{-2}/H^4, \quad \epsilon _t =\kappa (\Delta T)^2 \textit{Nu}/H^2, \end{eqnarray}

or in dimensionless form:

(3.13) \begin{eqnarray} D_\epsilon =\dfrac {\epsilon _k H}{U^3}=(\textit{Nu}-1) \textit{Ra} \textit{Pr}^{-2} {\textit{Re}}^{-3},\quad D_T=\frac {\epsilon _t H}{(\Delta T)^2 U}= \textit{Nu}\ {\textit{Re}}^{-1} \textit{Pr}^{-1}. \end{eqnarray}

Their behaviour as a function of ${\textit{Re}}$ or $Pe={\textit{Re}} \textit{Pr}$ is shown in figures 7(b) and 7(c). In both cases, they decay steadily, showing no evidence of saturation. Saturation of both $D_\epsilon$ and $D_T$ would correspond to the ultimate regime of convection, in which $\textit{Nu}\sim {\textit{Re}}\textit{Pr}\sim (\textit{Ra} \textit{Pr})^{1/2}$ .

The forcing being due to the buoyancy, we have $F_0=g\beta \Delta T$ . Therefore, the efficiency can be simply expressed as a function of the Rayleigh and Reynolds numbers as

(3.14) \begin{eqnarray} {\mathcal{E}}={\textit{Re}}^2 \textit{Pr}\textit{Ra}^{-1}. \end{eqnarray}

Injecting this into (3.13), we then get $D_\epsilon =(\textit{Nu}-1)/({\textit{Re}}\textit{Pr}{\mathcal{E}})$ . Because of the exact bound $\textit{Nu}\lt {\textit{Re}} \textit{Pr}$ (Lohse & Shishkina Reference Lohse and Shishkina2024), we see that $D_\epsilon$ is indeed bounded by $1/{\mathcal{E}}$ , in accordance with (2.8). This analysis therefore provides an alternative interpretation of the required conditions to get the ultimate, asymptotic regime, as being achieved when both $\textit{Nu}/({\textit{Re}}\textit{Pr})$ and ${\mathcal{E}}$ saturate. The verification of such saturation is provided in figure 8. We observe no sign of saturation for either quantity at the Rayleigh numbers investigated.

Figure 8. Tests of the ultimate regime in Rayleigh–Bénard convection. (a) Efficiency as a function of $\textit{Ra}$ . (b)  $\textit{Nu}/{\textit{Re}}\textit{Pr}$ as a function of $\textit{Ra}$ . Same symbols and colours as in figure 7. Data courtesy of O. Shishkina.

To investigate the inviscid limit of $D_\epsilon$ and ${\mathcal{E}}$ , one can use the extension of the Grossmann and Lohse theory to the ultimate regime (Lohse & Shishkina Reference Lohse and Shishkina2024). This theory identifies four different asymptotic regimes (for $ \textit{Ra}\to \infty$ ) depending on the scaling of $\textit{Pr}$ with $\textit{Ra}$ , namely:

1. (i) Regime $\textit{II}^{\prime}_l$ for $\textit{Pr}\lt \textit{Ra}^{-1}$ .

2. (ii) Regime $\textit{IV}^{\prime}_l$ for $\textit{Ra}^{-1}\lt \textit{Pr}\lt 1$ .

3. (iii) Regime $\textit{IV}^{\prime}_u$ for $1\lt \textit{Pr}\lt \textit{Ra}^{1/3}$ .

4. (iv) Regime $\textit{III}^{\prime}_\infty$ for $\textit{Ra}^{1/3}\lt \textit{Pr}\lt \textit{Ra}^{2/3}$ .

Table 3 summarises the behaviour of the Nusselt number, Reynolds number, dissipation and efficiency in these four regimes. We see that in regimes $\textit{IV}^{\prime}_l$ and $\textit{IV}^{\prime}_u$ , $D_\epsilon$ never achieves saturation at infinite Rayleigh number, at variance with the zeroth law of turbulence, while efficiency saturates to a constant independent of $\textit{Pr}$ . In contrast, in regime $\textit{II}^{\prime}_l$ , dissipation tends to a constant, while efficiency goes to zero. Finally, in regime $\textit{III}^{\prime}_\infty$ , the dissipation tends to zero and the efficiency to $\infty$ . Rayleigh–Bénard convection therefore provides an example where efficiency and energy dissipation can have different behaviours. Such difference could be used to offer an alternative detection of the ultimate regime.

Table 3. Variation of the Nusselt number $\textit{Nu}$ , the Reynolds number ${\textit{Re}}$ , the dimensionless dissipation $D_\epsilon$ and the efficiency ${\mathcal{E}}$ as a function of the Rayleigh number $\textit{Ra}$ and the Prandtl number $\textit{Pr}$ , in the four asymptotic ultimate regimes identified by Lohse & Shishkina (Reference Lohse and Shishkina2024). The last two columns are mere consequences of the definition of $D_\epsilon$ and ${\mathcal{E}}$ .