Katepalli R. Sreenivasan

Recent investigations of turbulent circulation fluctuations have uncovered substantial insights into the statistical organization of flow structures and revealed unexpected geometric features of turbulent intermittency. Of particular interest here is the observation that circulation probability distribution functions admit a superstatistical representation, namely a description based on "ensembles of Boltzmann"Gibbs ensembles." A fundamental phenomenological ingredient of this approach, which serves as a natural starting point for modeling, relies on the strong correlation between the dissipation field and the spatial distribution of elementary circulation-carrying structures, i.e., small-scale vortices. Within the language of superstatistics, this corresponds to characterizing circulation statistics through an appropriate choice of conditioned (Boltzmann-like) distributions and mixing distributions. We show that the superstatistical class of [Formula: see text]-exponentials, known to have broad applicability in a wide range of multiscale and nonequilibrium systems, provides an accurate description of the observed circulation statistics in homogeneous and isotropic turbulence. This finding opens avenues for exploring the statistical structure of the turbulent cascade in the context of nonextensive statistical mechanics, rooted in the concept of nonadditive entropies.

We present three-dimensional direct numerical simulations of turbulent Rayleigh–Bénard convection in a closed rectangular box whose width $L_y$ and length $L_x$ are 0.8 and 2.4 times the height $H$ , respectively. The Rayleigh number $\textit{Ra}$ varies from $10^5$ to $10^{10}$ , and the Prandtl number is unity. The advantages of the present configuration are: ( a ) a relatively stable unidirectional large-scale circulation, consisting of two counter-rotating rolls, fills the cell and fixes the thermal plume ejection- and shear-dominated regions, in contrast to those in closed cylindrical cells. ( b ) The regions of plume ejection are essentially independent of the sidewalls so that their autonomous existence can be studied. This is because there is some space, or ‘fetch’, for the velocity and thermal boundary layers to develop along the length. ( c ) This geometry allows one to study the influence of locally thin and thick boundary layers (which follow larger or smaller plume activity) on the scaling of convection properties. In regions of larger plume activity (defined by an incessant movement of plumes), the temperature fluctuation as well as the normalised thermal and viscous dissipation rates decay more slowly with $\textit{Ra}$ than in regions of lower activity. Both viscous and thermal boundary layers thin down rapidly with increasing distance from the plume ejection region. The local thicknesses of both boundary layers decline more rapidly with $\textit{Ra}$ in the ejection region than in regions of impact and shear, where they are similar to each other. Despite these details, the global heat transport laws are practically the same as those in other configurations of low to moderate aspect ratios.

To study the physics of small-scale properties of homogeneous isotropic turbulence at increasingly high Reynolds numbers, direct numerical simulation results have been obtained for forced isotropic turbulence at Taylor-scale Reynolds number $R_\lambda =2500$ on a $32\,768^3$ three-dimensional periodic domain using a GPU pseudo-spectral code on a 1.1 exaflop GPU supercomputer (Frontier). These simulations employ the multi-resolution independent simulation (MRIS) technique (Yeung & Ravikumar 2020, Phys. Rev. Fluids , vol. 5, 110517) where ensemble averaging is performed over multiple short segments initiated from velocity fields at modest resolution, and subsequently taken to higher resolution in both space and time. Reynolds numbers are increased by reducing the viscosity with the large-scale forcing parameters unchanged. Although MRIS segments at the highest resolution for each Reynolds number last for only a few Kolmogorov time scales, small-scale physics in the dissipation range is well captured – for instance, in the probability density functions and higher moments of the dissipation rate and enstrophy density, which appear to show monotonic trends persisting well beyond the Reynolds number range in prior works in the literature. Attainment of range of length and time scales consistent with classical scaling also reinforces the potential utility of the present high-resolution data for studies of short-time-scale turbulence physics at high Reynolds numbers where full-length simulations spanning many large-eddy time scales are still not accessible. A single snapshot of the $32\,768^3$ data is publicly available for further analyses via the Johns Hopkins Turbulence Database.

Experimental and numerical studies of incompressible turbulence suggest that the mean dissipation rate of kinetic energy remains constant as the Reynolds number tends to infinity (or the nondimensional viscosity tends to zero). This anomalous behavior is central to many theories of high-Reynolds-number turbulence and has been termed the "zeroth law" for this reason. Here, we report a sequence of direct numerical simulations of incompressible Navier-Stokes in a box with periodic boundary conditions, which indicate the likelihood that the anomaly vanishes at a rate that agrees with the scaling of third moment of absolute velocity increments. Our results suggest that turbulence without solid boundaries or walls may not develop strong enough singularities to sustain the strict version of the zeroth law.

The link between characteristic coherent structures and their statistical properties in turbulent flows remains largely unclear and is thus a central bottleneck for a better understanding of turbulent flows. Here, we demonstrate this link for the important problem of thermal convection. We show how the hierarchical plume network in the near-wall region of the flow, which becomes increasingly sparse with increasing distance away from the wall, is connected to the marginal stability of the thermal boundary layer and the resulting global heat transport. Our results, which are based on a series of direct numerical simulations for Rayleigh numbers up to [Formula: see text] in a relatively shallow layer, suggest a highly fluctuating thermal boundary layer that is composed of local building blocks in terms of plumes, which are the essential drivers of turbulent heat transport. These thermal plumes are found in a dynamically perpetual process of formation and aggregation that can be described, particularly well for Rayleigh numbers [Formula: see text], by a von Smoluchowski equation resulting in a gamma distribution of the local plume spacing, consistent with measurements. Similarity manifests with respect to the horizontal extension of the network, the vertical hierarchical plume clustering away from the wall and the number of plumes, over an order of magnitude of the thermal boundary layer thickness. Our findings suggest the dominance of dynamical local processes near the wall, rather than a global boundary layer instability.

This article presents my personal perspective on fluid mechanics research as it has evolved over the years, with a few specific references to the Division of Fluid Dynamics of the American Physical Society and its relation to the Physics of Fluids. The article is far from comprehensive, but includes some comments on this special collection.

We simulate thermal convection in a two-dimensional square box using the no-slip condition on all boundaries, and isothermal bottom and top walls, and adiabatic sidewalls. We choose 0.1 and 1 for the Prandtl number $Pr$ and vary the Rayleigh number $Ra$ between $10^6$ and $10^{12}$ . We particularly study the temporal evolution of integral transport quantities towards their steady states. Perhaps not surprisingly, the velocity field evolves more slowly than the thermal field, and its steady state – which is nominal in the sense that large-amplitude low-frequency oscillations persist around plausible averages – is reached exponentially. We study these oscillation characteristics. The transient time for the velocity field to achieve its nominal steady state increases almost linearly with the Reynolds number. For large $Ra$ , the Reynolds number itself scales almost as $Ra^{2/3}\, Pr^{-1}$ , and the Nusselt number as $Ra^{2/7}$ .

Intermittency of energy dissipation has long been studied via high-order moments in homogeneous and isotropic turbulence, but not much where the boundary effects are explicitly included. Here, we derive two fundamental Reynolds number scaling expressions for dissipation moments in wall-bounded flows -- one in the outer region where the boundary effects are weak and the other close to the walls where those effects are strong -- and support these expressions by direct numerical simulations. Dissipation moments in the outer region follow universal power laws with exponents linked to anomalous scaling of velocity structure functions. In contrast, moments near the wall follow a bounded defect law, leading to a finite asymptotic limit without intermittency. For very large Reynolds numbers, the outer proposal predicts vanishing dissipation compared to that on the wall, highlighting the need for solid boundaries in generating Onsager-type singularities.

Many claims of computational advantages have been made for quantum computing over classical but they have not been demonstrated for practical problems. Here, we present algorithms for solving time-dependent PDEs, with particular reference to fluid equations. We build on an idea based on a linear combination of unitaries to simulate nonunitary, non-Hermitian quantum systems, and generate hybrid quantum-classical algorithms that efficiently perform iterative matrix-vector multiplication and matrix inversion operations. These algorithms are end-to-end, with relatively low-depth quantum circuits that demonstrate quantum advantage, with the best-case asymptotic complexities, which we show are near optimal. We demonstrate the performance of the algorithms by conducting: (a) fully gate level, state-vector simulations using an in-house, high-performance, quantum simulator called ; (b) experiments on a real quantum device; and (c) noisy simulations using . We also provide device specifications such as error rates (noise) and state sampling (measurement) to accurately perform convergent flow simulations on noisy devices. The results offer evidence that the proposed algorithm is amenable for use on near-term quantum devices.

Roddam Narasimha’s scientific research in fluid dynamics encompassed turbulent flows—including the transition from the laminar state to turbulence and vice versa —kinetic theory of gases and structure of shock waves, nonlinear dynamics, aspects of tropical meteorology motivated by monsoons, as well as problems related to aeronautical engineering. Besides his life-long engagement in teaching and advanced research, he held leadership positions in learned societies and national institutions, he articulated original perspectives on Indic philosophy as well as history of science, and he provided diligent advice to the highest levels of government. In particular, he was deeply immersed in India’s technology development—specifically the light combat aircraft and the light transport aircraft—and worked on the implications of commercial supersonic flights over Indian airspace, stochastic theory of airworthiness of aircraft, etc. Above all, Roddam was known for the incisive quality, depth and perspective of his work, the dignity of personal interactions with people at all levels, and his energizing influence on younger colleagues.

Abstract Wall turbulence consists of various sizes of vortical structures that induce flow circulation around a wide range of closed Eulerian loops. Here we investigate the multiscale properties of circulation around such loops in statistically homogeneous planes parallel to the wall. Using a high-resolution direct numerical simulation database of turbulent channels at Reynolds numbers of $Re_\tau =180$ , 550, 1000 and 5200, circulation statistics are obtained in planes at different wall-normal heights. Intermittency of circulation in the planes of the outer flow ( $y^+ \gtrsim 0.1Re_\tau$ ) takes the form of universal bifractality as in homogeneous and isotropic turbulence. The bifractal character simplifies to space-filling character close to the wall, with scaling exponents that are linear in the moment order, and lower than those given by the Kolmogorov paradigm. The probability density functions of circulation are long-tailed in the outer bifractal region, with evidence showing their invariance with respect to the loop aspect ratio, while those in the inner region are closely Gaussian. The unifractality near the wall implies that the circulation there is not intermittent in character.

Abstract Solar inertial modes have the potential to surpass the diagnostic capabilities of acoustic waves in probing the deep interior of the Sun. The fulfillment of this potential requires an accurate identification and characterization of these modes. Among the set of detected inertial modes, the equatorially antisymmetric “high-frequency retrograde” (HFR) modes has attracted special interest because numerical studies have suggested that they are not purely toroidal, as initial observations suggested, and predicted that they would possess a significant radial flow signal at depth. Here, we analyze ∼13 yr of Helioseismic and Magnetic Imager/Solar Dynamics Observatory 5° ring tiles and discover a horizontal-divergence signal, directly connected to radial flows, in the near-surface layers of the Sun. We demonstrate that this signal is indeed part of the HFR modes and not spatial leakage from prograde flows associated with magnetic regions. The amplitudes of the horizontal divergence are approximately half that associated with radial vorticity. We also report the presence of a ridge of enhanced power, although with a signal-to-noise ratio of 0.3, in the retrograde frequencies that coincides with the HFR latitudinal overtones reported by models. Using numerical linear models, we find reasonable agreement with observations, though future work on boundary considerations and the inclusion of the near-surface may improve future inferences. This is the first instance where numerical studies of solar inertial modes have guided observations, giving further confidence to past inferences that rely upon numerical models.

Understanding the effects of solid boundaries on turbulent fluctuations remains a long-standing challenge. Available data on mean-square fluctuations in these flows show apparent contradiction with classical scaling. We had earlier proposed an alternative model based on the principle of bounded dissipation. Despite its putative success, a conclusive outcome requires much higher Reynolds numbers than are available at present, or can be expected to be available in the near future. However, the model can be validated satisfactorily even within the Reynolds number range already available by considering high-order moments and their distributions in the wall-normal direction. Expressions for high-order moments of streamwise velocity fluctuation [Formula: see text] are derived in the form [Formula: see text], where the superscript [Formula: see text] indicates the wall unit normalization, and brackets stand for averages over time and the homogeneous plane normal to the wall, [Formula: see text] is an integer, [Formula: see text] and [Formula: see text] are constants independent of the friction Reynolds number [Formula: see text], and [Formula: see text] is the distance away from the wall, normalized by the flow thickness [Formula: see text]. In particular, [Formula: see text] according to the "linear q-norm Gaussian" process, where [Formula: see text] and [Formula: see text] are flow-independent constants. Excellent agreement is found between this formula and the available data in boundary layers, pipes, and channels for [Formula: see text]. For fixed [Formula: see text], the present formulation leads to the bounded state [Formula: see text] as [Formula: see text]. This work demonstrates the success of the present model in describing the behavior of fluctuations in wall flows.

The applications and impact of high fidelity simulation of fluid flows are far-reaching. They include settling some long-standing and fundamental questions in turbulence. However, the computational resources required for such efforts are extensive. Here, we explore the possibility of employing the recent computing paradigm of quantum computing to simulate fluid flows. The lure of this new paradigm is the potentially exponential advantage in memory and speed, in comparison with classical computing. This field has recently witnessed a considerable uptick in excitement and contributions. In this work, we give a succinct discussion of the progress made so far, with focus on fluid flows, accompanied by an enumeration of challenges that require sustained efforts for progress. Quantum computing of fluid flows has a promising future, but the inherently nonlinear nature of flows requires serious efforts on resolving various bottlenecks, and on synthesising progress on theoretical, numerical and experimental fronts. We present certain critical details that have not yet attracted adequate attention.

Solar inertial modes have the potential to surpass the diagnostic capabilities of acoustic waves in probing the deep interior of the Sun. The fulfillment of this potential requires an accurate identification and characterization of these modes. Among the set of detected inertial modes, the equatorially anti-symmetric "high-frequency retrograde'' (HFR) modes has attracted special interest because numerical studies have suggested that they are not purely toroidal, as initial observations suggested, and predicted that they would possess a significant radial flow signal at depth. Here, we analyze $\sim$13 years of HMI/SDO 5$^\circ$ ring tiles, and discover a horizontal-divergence signal, directly connected to radial flows, in the near surface layers of the Sun. We demonstrate that this signal is indeed part of the HFR modes and not spatial leakage from prograde flows associated with magnetic regions. The amplitudes of the horizontal divergence are approximately half that associated with radial vorticity. We also report the presence of a ridge of enhanced power, although with a signal-to-noise ratio of 0.3, in the retrograde frequencies that coincides with the HFR latitudinal overtones reported by models. Using numerical linear models we find reasonable agreement with observations, though future work on boundary considerations and the inclusion of the near-surface may improve future inferences. This is the first instance where numerical studies of solar inertial modes have guided observations, giving further confidence to past inferences that rely upon numerical models.

Turbulent motion of fluids is often thought of as a grand problem, but what exactly is this “turbulence problem”? Because it has often been proclaimed as very difficult and unsolved, when can we claim that it is solved? How does this situation in turbulence compare with other complex problems in physical sciences? Addressing these questions is not trivial because everyone has their favorite idea of what is required of the “solution.” The answers range from being able to calculate the pressure drop in turbulent pipe flow to being able to calculate anomalous scaling exponents to answering the regularity problem of the Navier–Stokes equations. Taking an absolute position on the basis of any of these, or other similar examples, is incomplete at best and potentially erroneous at worst. We believe that it is beneficial to have an open discussion of this topic for the advancement of the research agenda in turbulence. This article is an attempt to address the question of what constitutes the turbulence problem, its place in the scientific enterprise as a whole, and how and when one may declare it as solved.

Wall turbulence consists of various sizes of vortical structures that induce flow circulation around a wide range of closed Eulerian loops. Here we investigate the multiscale properties of circulation around such loops in statistically homogeneous planes parallel to the wall. Using a high-resolution direct numerical simulation database of turbulent channels at Reynolds numbers of $Re_τ=180$, 550, 1000 and 5200, circulation statistics are obtained in planes at different wall-normal heights. Intermittency of circulation in the planes of the outer flow ($y^+ \gtrsim 0.1Re_τ$) takes the form of universal bifractality as in homogeneous and isotropic turbulence. The bifractal character simplifies to space-filling character close to the wall, with scaling exponents that are linear in the moment order, and lower than those given by the Kolmogorov paradigm. The probability density functions of circulation are long-tailed in the outer bifractal region, {with evidence showing their invariance with respect to the loop aspect ratio}, while those in the inner region are closely Gaussian. The unifractality near the wall implies that the circulation there is not intermittent in character.

The link between characteristic coherent structures and their statistical properties in turbulent flows remains largely unclear and is thus a central bottleneck for a better understanding of turbulent flows. Here, we demonstrate this link for the important problem of thermal convection. We show how the hierarchical plume network in the near-wall region of the flow, which becomes increasingly sparse with increasing distance away from the wall, is connected to the marginal stability of the thermal boundary layer and the resulting global heat transport. Our results, which are based on a series of direct numerical simulations for Rayleigh numbers up to $Ra=10^{11}$ in a relatively shallow layer, suggest a highly fluctuating thermal boundary layer that is composed of local building blocks in terms of plumes, which are the essential drivers of turbulent heat transport. These thermal plumes are found in a dynamically perpetual process of formation and aggregation that can be described, particularly well for Rayleigh numbers $Ra ≳ 10^9$, by a von Smoluchowski equation resulting in a gamma distribution of the local plume spacing, consistent with measurements. Similarity manifests with respect to the horizontal extension of the network, the vertical hierarchical plume clustering away from the wall and the number of plumes, over an order of magnitude of the thermal boundary layer thickness. Our findings suggest the dominance of dynamical local processes near the wall, rather than a global boundary layer instability.

Turbulent convection in the interiors of the Sun and the Earth occurs at high Rayleigh numbers $Ra$ , low Prandtl numbers $Pr$ , and different levels of rotation rates. To understand the combined effects better, we study rotating turbulent convection for $Pr = 0.021$ (for which some laboratory data corresponding to liquid metals are available), and varying Rossby numbers $Ro$ , using direct numerical simulations in a slender cylinder of aspect ratio 0.1; this confinement allows us to attain high enough Rayleigh numbers. We are motivated by the earlier finding in the absence of rotation that heat transport at high enough $Ra$ is similar between confined and extended domains. We make comparisons with higher aspect ratio data where possible. We study the effects of rotation on the global transport of heat and momentum as well as flow structures ( a ) for increasing rotation at a few fixed values of $Ra$ , and ( b ) for increasing $Ra$ (up to $10^{10}$ ) at the fixed, low Ekman number $1.45 \times 10^{-6}$ . We compare the results with those from unity $Pr$ simulations for the same range of $Ra$ and $Ro$ , and with the non-rotating case over the same range of $Ra$ and low $Pr$ . We find that the effects of rotation diminish with increasing $Ra$ . These results and comparison studies suggest that for high enough $Ra$ , rotation alters convective flows in a similar manner for small and large aspect ratios, so useful insights on the effects of high thermal forcing on convection can be obtained by considering slender domains.

Modeling heat transport by convection is one of the most challenging aspects of solar and stellar physics. The literature currently provides apparently inconsistent observational estimates of the strength of large-scale convective flows in the upper layers of the solar convection zone. In addition, the large-scale convective flows predicted from numerical simulations are substantially stronger than some of the observational inferences in the literature. The current work aims to provide a consistent presentation of some of the main results in the literature both from observations and simulations. To achieve this aim, we carry out an analysis of published estimates of the strength of solar convection at different spatial scales. In particular, we employ a consistent set of conventions to compute the kinetic energy density in the east-west flows. This establishes a clear baseline for future work. The main conclusion is that there are inconsistencies between different observational results and also differences between observations and simulations. This conclusion is important as it demonstrates a need to determine the sources of the inconsistencies between different observational inferences and also to determine the missing ingredients in simulations of solar subsurface convection.

We study the dynamics of thermal and momentum boundary regions in three-dimensional direct numerical simulations of Rayleigh–Bénard convection for the Rayleigh-number range $10^5\leq Ra \leq 10^{11}$ and $Pr=0.7$ . Using a Cartesian slab with horizontal periodic boundary conditions and an aspect ratio of 4, we obtain statistical homogeneity in the horizontal $x$ - and $y$ -directions, thus approximating best an extended convection layer relevant for most geo- and astrophysical flow applications. We observe upon canonical use of combined long-time and area averages, with averaging periods of at least 100 free-fall times, that a global coherent mean flow is practically absent and that the magnitude of the velocity fluctuations is larger than the mean by up to 2 orders of magnitude. The velocity field close to the wall is a collection of differently oriented local shear-dominated flow patches interspersed by extensive shear-free incoherent regions which can be as large as the whole cross-section, unlike for a closed cylindrical convection cell of aspect ratio of the order 1. The incoherent regions occupy a 60 % area fraction for all Rayleigh numbers investigated here. Rather than resulting in a pronounced mean flow with small fluctuations about such a mean, as found in small-aspect-ratio convection, the velocity field is dominated by strong fluctuations of all three components around a non-existent or weak mean. We discuss the consequences of these observations for convection layers with larger aspect ratios, including boundary layer instabilities and the resulting turbulent heat transport.

Abstract A direct dynamical test of the sunspot cycle is carried out to indicate that a stochastically forced nonlinear oscillator characterizes its dynamics. The sunspot series is then decomposed into its eigen time-delay coordinates. The relevant analysis reveals that the sunspot series exhibits bistability, with the possibility of modeling the solar cycle as a stochastically and periodically forced bistable oscillator, accounting for poloidal and toroidal modes of the solar magnetic field. Such a representation enables us to conjecture stochastic resonance as the key mechanism in amplifying the planetary influence on the Sun, and that extreme events, due to turbulent convection noise inside the Sun, dictate crucial phases of the sunspot cycle, such as the Maunder minimum.

The applications and impact of high fidelity simulation of fluid flows are far-reaching. They include settling some long-standing and fundamental questions in turbulence. However, the computational resources required for such efforts are extensive. Here, we explore the possibility of employing the recent computing paradigm of quantum computing to simulate fluid flows. The lure of this new paradigm is the potentially exponential advantage in memory and speed, in comparison with classical computing. This field has recently witnessed a considerable uptick in excitement and contributions. In this work, we give a succinct discussion of the progress made so far, with focus on fluid flows, accompanied by an enumeration of challenges that require sustained efforts for progress. Quantum computing of fluid flows has a promising future, but the inherently nonlinear nature of flows requires serious efforts on resolving various bottlenecks, and on synthesising progress on theoretical, numerical and experimental fronts. We present certain critical details that have not yet attracted adequate attention.

A recent discovery about the inertial range of homogeneous and isotropic turbulence is the saturation of the scaling exponents <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:msub><a:mi>ζ</a:mi><a:mi>n</a:mi></a:msub></a:math> for large <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mi>n</b:mi></b:math>, defined via structure functions of order <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mi>n</c:mi></c:math> as <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:mrow><d:msub><d:mi>S</d:mi><d:mi>n</d:mi></d:msub><d:mrow><d:mo>(</d:mo><d:mi>r</d:mi><d:mo>)</d:mo></d:mrow><d:mo>=</d:mo><d:mrow><d:mo>〈</d:mo><d:msup><d:mrow><d:mo>(</d:mo><d:msub><d:mi>δ</d:mi><d:mi>r</d:mi></d:msub><d:mi>u</d:mi><d:mo>)</d:mo></d:mrow><d:mi>n</d:mi></d:msup><d:mo>〉</d:mo></d:mrow><d:mo>=</d:mo><d:mi>A</d:mi><d:mrow><d:mo>(</d:mo><d:mi>n</d:mi><d:mo>)</d:mo></d:mrow><d:msup><d:mi>r</d:mi><d:msub><d:mi>ζ</d:mi><d:mi>n</d:mi></d:msub></d:msup></d:mrow></d:math>. We focus on longitudinal structure functions for <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"><e:mrow><e:msub><e:mi>δ</e:mi><e:mi>r</e:mi></e:msub><e:mi>u</e:mi></e:mrow></e:math> between two positions that are <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"><f:mi>r</f:mi></f:math> apart in the same direction as <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"><g:mrow><g:mi>u</g:mi></g:mrow></g:math>. In a previous work [], two of the present authors developed a theory for <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"><h:msub><h:mi>ζ</h:mi><h:mi>n</h:mi></h:msub></h:math>, which agrees with measurements for all <i:math xmlns:i="http://www.w3.org/1998/Math/MathML"><i:mi>n</i:mi></i:math> for which reliable data are available, and shows saturation for large <j:math xmlns:j="http://www.w3.org/1998/Math/MathML"><j:mi>n</j:mi></j:math>. Here, we derive expressions for the probability density functions of <k:math xmlns:k="http://www.w3.org/1998/Math/MathML"><k:mrow><k:msub><k:mi>δ</k:mi><k:mi>r</k:mi></k:msub><k:mi>u</k:mi></k:mrow></k:math> for four different states of turbulence, including the asymptotic fourth state defined by the saturation of exponents for large <l:math xmlns:l="http://www.w3.org/1998/Math/MathML"><l:mi>n</l:mi></l:math>. This saturation means that the scale separation is violated in favor of strongly coupled quasiordered flow structures, which likely take the form of long and thin (worm-like) structures of length <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mi>L</m:mi></m:math> and thickness <n:math xmlns:n="http://www.w3.org/1998/Math/MathML"><n:mrow><n:mi>l</n:mi><n:mo>=</n:mo><n:mi>O</n:mi><n:mo>(</n:mo><n:mi>L</n:mi><n:mo>/</n:mo><n:mi>R</n:mi><n:mi>e</n:mi><n:mo>)</n:mo></n:mrow></n:math>. Published by the American Physical Society 2024

Two quantum algorithms are presented for the numerical solution of a linear one-dimensional advection–diffusion equation with periodic boundary conditions. Their accuracy and performance with increasing qubit number are compared point-by-point with each other. Specifically, we solve the linear partial differential equation with a Quantum Linear Systems Algorithm (QLSA) based on the Harrow–Hassidim–Lloyd method and a Variational Quantum Algorithm (VQA), for resolutions that can be encoded using up to 6 qubits, which corresponds to N=64 grid points on the unit interval. Both algorithms are hybrid in nature, i.e., they involve a combination of classical and quantum computing building blocks. The QLSA and VQA are solved as ideal statevector simulations using the in-house solver QFlowS and open-access Qiskit software, respectively. We discuss several aspects of both algorithms which are crucial for a successful performance in both cases. These are the accurate eigenvalue estimation with the quantum phase estimation for the QLSA and the choice of the algorithm of the minimization of the cost function for the VQA. The latter algorithm is also implemented in the noisy Qiskit framework including measurement noise. We reflect on the current limitations and suggest some possible routes of future research for the numerical simulation of classical fluid flows on a quantum computer.

Turbulent wall flows offer the most direct means for understanding the effects of boundaries and viscosity on turbulent fluctuations. Available data on mean-square fluctuations in these flows show apparent contradiction with classical scaling based on the {mean} wall shear stress. We had earlier proposed an alternative model based on the principle of bounded dissipation to describe the data. Despite its putative success, a conclusive outcome requires much higher Reynolds numbers than are available at present, or can be expected to be available in the near future. However, the model can be validated satisfactorily even within the Reynolds number range already available by considering high-order moments and their distributions in the wall-normal direction. Expressions for high-order moments of streamwise velocity fluctuation $u$ are derived in the form $ \langle u^{+2q} \rangle^{1/q}=α_q-β_q y^{\ast1/4}$; here $q$ is an integer, $α_q$ and $β_q$ are constants independent of the friction Reynolds number $Re_τ$, and $y^{\ast} = y/δ$ is the distance away from the wall, normalized by the flow thickness $δ$; in particular, $α_q =μ+σq$ according to the `linear q-norm Gaussian' process, where $μ$ and $σ$ are flow-independent constants. Excellent agreement is found between this formula and the available data in boundary layers, pipes and channels for $1 \leq q \leq 5$. For fixed $y^+ = y^*Re_τ$, the present formulation leads to the bounded state $\langle u^{+2q} \rangle^{1/q}=α_q$ as $Re_τ\rightarrow\infty$. This work demonstrates the success of the present model in describing the behavior of fluctuations in wall flows.

A direct dynamical test of the sunspot-cycle is carried out which indicates that a stochastically forced non-linear oscillator characterizes its dynamics. The sunspot series is then decomposed into its eigen time-delay coordinates. The analysis of these coordinates reveals that the sunspot series exhibits bistability, and suggests the possibility of modeling the solar cycle as a stochastically and periodically forced bistable oscillator, accounting for the Poloidal and Toroidal modes of the solar magnetic field. Such a representation of the sunspot series in terms of stochastic bistable dynamical system enables us to conjecture stochastic resonance as the key mechanism in amplifying the planetary influence of Jupiter on the sun, and that extreme events, due to turbulent convection noise inside the sun, dictate crucial phases of the sunspot cycle, such as the Maunder minimum.

Abstract Inertial wave modes in the Sun are of interest owing to their potential to reveal new insight into the solar interior. These predominantly retrograde-propagating modes in the solar subsurface appear to deviate from the thin-shell Rossby–Haurwitz model at high azimuthal orders. We present new measurements of sectoral inertial modes at m &gt; 15 where the modes appear to become progressively less retrograde compared to the canonical Rossby–Haurwitz dispersion relation in a corotating frame. We use a spectral eigenvalue solver to compute the spectrum of solar inertial modes in the presence of differential rotation. Focussing specifically on equatorial Rossby modes, we find that the numerically obtained mode frequencies lie along distinct ridges, one of which lies strikingly close to the observed mode frequencies in the Sun. We also find that the n = 0 ridge is deflected strongly in the retrograde direction. This suggests that the solar measurements may not correspond to the fundamental n = 0 Rossby–Haurwitz solutions as was initially suspected, but to those for a higher n . The numerically obtained eigenfunctions also appear to sit deep within the convection zone—unlike those for the n = 0 modes—which differs substantially from solar measurements and complicates inference.

We study the dynamics of thermal and momentum boundary regions in three-dimensional direct numerical simulations of Rayleigh-Bénard convection for the Rayleigh number range $10^5 \le Ra \le 10^{11}$ and $Pr=0.7$. Using a Cartesian slab with horizontal periodic boundary conditions and an aspect ratio of 4, we obtain statistical homogeneity in the horizontal $x$- and $y$-directions, thus approximating best an extended convection layer relevant for most geo- and astrophysical flow applications. We observe upon canonical use of combined long-time and area averages, with averaging periods of at least 100 free-fall times, that a global coherent mean flow is practically absent and that the magnitude of the velocity fluctuations is larger than the mean by up to 2 orders of magnitude. The velocity field close to the wall is a collection of differently oriented local shear-dominated flow patches interspersed by extensive shear-free incoherent regions which can be as large as the whole cross section, unlike for a closed cylindrical convection cell of aspect ratio of the order 1. The incoherent regions occupy a $60\%$ area fraction for all Rayleigh numbers investigated here. Rather than resulting in a pronounced mean with small fluctuations about such a mean, as found in small-aspect-ratio convection, the velocity field is dominated by strong fluctuations of all three components around a non-existent or weak mean. We discuss the consequences of these observations for convection layers with larger aspect ratios, including boundary layer instabilities and the resulting turbulent heat transport.

Transition from laminar to turbulent states of classical viscous fluids is complex and incompletely understood. Transition to quantum turbulence (QT), by which we mean the turbulent motion of quantum fluids such as helium II, whose physical properties depend on quantum physics in some crucial respects, is naturally more complex. This increased complexity arises from superfluidity, quantization of circulation, and, at finite temperatures below the critical, the two-fluid behavior. Transition to QT could involve, as an initial step, the transition of the classical component, or the intrinsic or extrinsic nucleation of quantized vortices in the superfluid component, or a simultaneous occurrence of both scenarios-and the subsequent interconnected evolution. In spite of the multiplicity of scenarios, aspects of transition to QT can be understood at a phenomenological level on the basis of some general principles, and compared meaningfully with transition in classical flows.

We provide a theoretical estimation of the relaxation time for the kinetic theoretical representation of homogeneous, isotropic, and stationary turbulent flow. The basic approach depends on the construction of a balance of fluctuation and dissipation between the relaxation process and the self-consistently generated fluctuating force. The kinetic theory representation generalizes the modeling of turbulence so that turbulent viscosity as well as all higher order transport coefficients are determined by the relaxation time. The resulting value of the relaxation time gives an estimation of the turbulent viscosity with the coefficient value comparable with those from most of the representative turbulence models, which have been obtained empirically or semi-empirically.

We present a full quantum algorithm for the lattice Boltzmann method for simulating fluid flows, the only such algorithm to implement both the streaming and collision substeps as unitary operators using an efficient number of qubits. We use Hamiltonian simulation as the main route underlying the algorithm, and show that it uses qubits that scale logarithmically in the flow Reynolds number but gates that scale only polynomially. We motivate the discussion by a brief overview of existing attempts at simulating classical fluids on quantum computers and present a pedagogical discussion on assigning quantum operators to classical variables in both the streaming and the collision substeps, after highlighting the incompatibility of the latter with the implementations of the streaming step that exists in the literature. We use the Bhatnagar–Gross–Krook ansatz for the collision term, representing the relaxation toward an equilibrium distribution. For nonlinear collisions, we use Kowalski's framework that links the nonlinear dynamics of a system to the evolution of bosonic modes, assigning a Carleman linearization order to the truncation in the Fock space of the bosons. We present the qubit and gate complexities, in terms of the chosen accuracy and the Reynolds number. In the Appendix, we work out the details of implementing the operators of the truncated bosonic Fock space in terms of single-qubit gates as well as the error scaling for a general polynomial driving function.

Two quantum algorithms are presented for the numerical solution of a linear one-dimensional advection-diffusion equation with periodic boundary conditions. Their accuracy and performance with increasing qubit number are compared point-by-point with each other. Specifically, we solve the linear partial differential equation with a Quantum Linear Systems Algorithms (QLSA) based on the Harrow--Hassidim--Lloyd method and a Variational Quantum Algorithm (VQA), for resolutions that can be encoded using up to 6 qubits, which corresponds to $N=64$ grid points on the unit interval. Both algorithms are of hybrid nature, i.e., they involve a combination of classical and quantum computing building blocks. The QLSA and VQA are solved as ideal statevector simulations using the in-house solver QFlowS and open-access Qiskit software, respectively. We discuss several aspects of both algorithms which are crucial for a successful performance in both cases. These are the sizes of an additional quantum register for the quantum phase estimation for the QLSA and the choice of the algorithm of the minimization of the cost function for the VQA. The latter algorithm is also implemented in the noisy Qiskit framework including measurement and decoherence circuit noise. We reflect the current limitations and suggest some possible routes of future research for the numerical simulation of classical fluid flows on a quantum computer.

We discuss the viability of ensemble simulations of fluid flows on quantum computers. The basic idea is to formulate a functional Liouville equation for the probability distribution of the flow field configuration and recognize that, due to its linearity, such an equation is in principle more amenable to quantum computing than the equations of fluid motion. After suitable marginalization and associated closure, the Liouville approach is shown to require several hundreds of logical qubits, hence calling for a major thrust in current noise correction and mitigation techniques.

In Nagib, Chauhan and Monkewitz~\cite{NCM07} we concluded that nearly all available $C_f$ relations for zero-pressure-gradient boundary layers are in remarkable agreement over the entire range $Re_θ$ $&lt;$ O($10^8$), provided one coefficient is adjusted in each relation by anchoring it to accurate measurements. Regarding the peak of the streamwise turbulence intensity $^+_P$, we conclude here that accurate measurements in flows with $Re_τ$ $&gt;$ O($10^6$) are required, especially when looking only at the peak $^+_P$ to discriminate between recently proposed trends. We also find remarkable agreement between the three analyses of Monkewitz \cite{M22}, Chen and Sreenivasan \cite{CS22} and Monkewitz and Nagib \cite{MN15}, with some coefficients slightly modified, by underpinning them with the same accurate measurements of $^+_P$ from reliable channel and boundary layer data. All the three analyses conclude that the inner peak of $^+$ remains finite in the limit of infinite Reynolds number, which is at variance with the unlimited growth of $^+_P$ as $\ln{Re}$ $_τ$ predicted by the attached eddy model \cite{MM19}. Accurate measurements of high-order moments and the guidance of consistent asymptotic expansions may help clarify the issue at lower $Re_τ$ values.

For quantum computing (QC) to emerge as a practically indispensable computational tool, there is a need for quantum protocols with end-to-end practical applications-in this instance, fluid dynamics. We debut here a high-performance quantum simulator which we term QFlowS (Quantum Flow Simulator), designed for fluid flow simulations using QC. Solving nonlinear flows by QC generally proceeds by solving an equivalent infinite dimensional linear system as a result of linear embedding. Thus, we first choose to simulate two well-known flows using QFlowS and demonstrate a previously unseen, full gate-level implementation of a hybrid and high precision Quantum Linear Systems Algorithms (QLSA) for simulating such flows at low Reynolds numbers. The utility of this simulator is demonstrated by extracting error estimates and power law scaling that relates [Formula: see text] (a parameter crucial to Hamiltonian simulations) to the condition number [Formula: see text] of the simulation matrix and allows the prediction of an optimal scaling parameter for accurate eigenvalue estimation. Further, we include two speedup preserving algorithms for a) the functional form or sparse quantum state preparation and b) in situ quantum postprocessing tool for computing nonlinear functions of the velocity field. We choose the viscous dissipation rate as an example, for which the end-to-end complexity is shown to be [Formula: see text], where [Formula: see text] is the size of the linear system of equations, [Formula: see text] is the solution error, and [Formula: see text] is the error in postprocessing. This work suggests a path toward quantum simulation of fluid flows and highlights the special considerations needed at the gate-level implementation of QC.

In continuation of our earlier work (Chen &amp; Sreenivasan, J. Fluid Mech. , vol. 908, 2021, R3; Chen &amp; Sreenivasan, J. Fluid Mech. , vol. 933, 2022 a , A20 – together referred to as CS hereafter), we present a self-consistent Reynolds number asymptotics for wall-normal profiles of variances of streamwise and spanwise velocity fluctuations as well as root-mean-square pressure, across the entire flow region of channel and pipe flows and flat-plate boundary layers. It is first shown that, when normalized by peak values, the Reynolds number dependence and wall-normal variation of all three profiles can be decoupled, in excellent agreement with available data, sharing the common inner expansion of the type $\phi (y^+)=f_0(y^+)+f_1(y^+)/Re^{1/4}_\tau$ , where $\phi$ is one of the quantities just mentioned, the functions $f_0$ and $f_1$ depend only on $y^+$ , and $Re_\tau$ is the friction Reynolds number. Here, the superscript $+$ indicates normalization by wall variables. We show that this result is completely consistent with CS. Secondly, by matching the above inner expansion and the outer flow similarity form, a bounded variation $\phi (y^\ast )=\alpha _\phi -\beta _{\phi }y^{{\ast {1}/{4}}}$ is derived for the outer region where, for each $\phi$ , the constants $\alpha _\phi$ and $\beta _{\phi }$ are independent of $Re_\tau$ and $y^\ast$ $\equiv y^+/Re_\tau$ – also in excellent agreement with simulations and experimental data. One of the predictions of the analysis is that, for asymptotically high Reynolds numbers, a finite plateau $\phi \approx \alpha _\phi$ appears in the outer region. This result sheds light on the intriguing issue of the outer shoulder of the variance of the streamwise velocity fluctuation, which should be bounded by the asymptotic plateau of approximately 10.