Growth Of Enstrophy Research Articles

The Dissipation of the Kinetic Energy for 2D Bounded Flow by Using Moment-Based Boundary Conditions with Burnett Order Stress for LBM

In this article, the lattice Boltzmann method with two relaxation time (TRT) for the D2Q9 model is used to investigate numerical results for 2D flow. The problem is performed to show the dissipation of the kinetic energy rate and its relationship with the enstrophy growth for 2D dipole wall collision. The investigation is carried out for normal collision and oblique incidents at an angle of . We prove the accuracy of moment -based boundary conditions with slip and Navier-Maxwell slip conditions to simulate this flow. These conditions are under the effect of Burnett-order stress conditions that are consistent with the discrete Boltzmann equation. Stable results are found by using this kind of boundary condition where dissipation of the kinetic energy is found to be proportional to in the first regime and it is in the second part of the regime as expected. An excellent agreement with the benchmark data is observed.

Alignments of triad phases in extreme one-dimensional Burgers flows.

We analyze the fine structure of nonlinear modal interactions in inviscid and viscous Burgers flows in 1D, which serve as toy models for the Euler and Navier-Stokes dynamics. This analysis is focused on preferential alignments characterizing the phases of Fourier modes participating in triadic interactions, which are key to determining the nature of energy fluxes between different scales. We develop diagnostic tools designed to probe the level of coherence among triadic interactions and apply them to Burgers flows corresponding to different initial conditions, including unimodal, extreme (in the sense of maximizing the growth of enstrophy in finite time), and generic. We find that in all cases triads involving energy-containing Fourier modes align their phases so as to maximize the energy flux toward small scales, and most of this flux is realized by only a handful of triads revealing a universal statistical distribution. We then identify individual triads making the largest contributions to the flux at different wave numbers and show that they represent a mixture of local and nonlocal interactions, with the latter becoming dominant at later times. These results point to the possibility of constructing a strongly reduced modal representation of Burgers flows that would require a much smaller number of degrees of freedom. Another interesting observation is that removing the spatial coherence from the extreme initial data (by randomizing the phases while retaining the magnitudes of the Fourier coefficients) does not profoundly change the nature of triadic interactions and synchronization as well as the resulting fluxes in these flows.

Sharp bounds on enstrophy growth for viscous scalar conservation laws

We prove sharp bounds on the enstrophy growth in viscous scalar conservation laws. The upper bound is, up to a prefactor, the enstrophy created by the steepest viscous shock admissible by the L∞ and total variation bounds and viscosity. This answers a conjecture by Ayala and Protas (2011 Physica D 240 1553–63), based on numerical evidence, for the viscous Burgers equation.

Potentially Singular Behavior of the 3D Navier–Stokes Equations

Whether the 3D incompressible Navier–Stokes equations can develop a finite time singularity from smooth initial data is one of the most challenging problems in nonlinear PDEs. In this paper, we present some new numerical evidence that the incompressible axisymmetric Navier–Stokes equations with smooth initial data of finite energy seem to develop potentially singular behavior at the origin. This potentially singular behavior is induced by a potential finite time singularity of the 3D Euler equations that we reported in a companion paper published in the same issue, see also Hou (Potential singularity of the 3D Euler equations in the interior domain. arXiv:2107.05870 [math.AP], 2021). We present numerical evidence that the 3D Navier–Stokes equations develop nearly self-similar singular scaling properties with maximum vorticity increased by a factor of \(10^7\). We have applied several blow-up criteria to study the potentially singular behavior of the Navier–Stokes equations. The Beale–Kato–Majda blow-up criterion and the blow-up criteria based on the growth of enstrophy and negative pressure seem to imply that the Navier–Stokes equations using our initial data develop a potential finite time singularity. We have also examined the Ladyzhenskaya–Prodi–Serrin regularity criteria (Kiselev and Ladyzhenskaya in Izv Akad Nauk SSSR Ser Mat 21(5):655–690, 1957; Prodi in Ann Math Pura Appl 4(48):173–182, 1959; Serrin in Arch Ration Mech Anal 9:187–191, 1962) that are based on the growth rate of \(L_t^q L_x^p\) norm of the velocity with \(3/p + 2/q \le 1\). Our numerical results for the cases of \((p,q) = (4,8),\; (6,4),\; (9,3)\) and \((p,q)=(\infty ,2)\) provide strong evidence for the potentially singular behavior of the Navier–Stokes equations. The critical case of \((p,q)=(3,\infty )\) is more difficult to verify numerically due to the extremely slow growth rate in the \(L^3\) norm of the velocity field and the significant contribution from the far field where we have a relatively coarse grid. Our numerical study shows that while the global \(L^3\) norm of the velocity grows very slowly, the localized version of the \(L^3\) norm of the velocity experiences rapid dynamic growth relative to the localized \(L^3\) norm of the initial velocity. This provides further evidence for the potentially singular behavior of the Navier–Stokes equations.

Searching for Singularities in Navier–Stokes Flows Based on the Ladyzhenskaya–Prodi–Serrin Conditions

In this investigation we perform a systematic computational search for potential singularities in 3D Navier-Stokes flows based on the Ladyzhenskaya-Prodi-Serrin conditions. They assert that if the quantity $\int_0^T \| \mathbf{u}(t) \|_{L^q(\Omega)}^p \, dt$, where $2/p+3/q \le 1$, $q > 3$, is bounded, then the solution $\mathbf{u}(t)$ of the Navier-Stokes system is smooth on the interval $[0,T]$. In other words, if a singularity should occur at some time $t \in [0,T]$, then this quantity must be unbounded. We have probed this condition by studying a family of variational PDE optimization problems where initial conditions $\mathbf{u}_0$ are sought to maximize $\int_0^T \| \mathbf{u}(t) \|_{L^4(\Omega)}^8 \, dt$ for different $T$ subject to suitable constraints. These problems are solved numerically using a large-scale adjoint-based gradient approach. Even in the flows corresponding to the optimal initial conditions determined in this way no evidence has been found for singularity formation, which would be manifested by unbounded growth of $\| \mathbf{u}(t) \|_{L^4(\Omega)}$. However, the maximum enstrophy attained in these extreme flows scales in proportion to $\mathcal{E}_0^{3/2}$, the same as found by Kang et al. (2020) when maximizing the finite-time growth of enstrophy. In addition, we also consider sharpness of an a priori estimate on the time evolution of $\| \mathbf{u}(t) \|_{L^4(\Omega)}$ by solving another PDE optimization problem and demonstrate that the upper bound in this estimate could be improved.

A novel compressible enstrophy transport equation-based analysis of instability during Magnus–Robins effects for high rotation rates

The effects of compressibility on the instability of a two-dimensional flow past a rotating cylinder executing high rotation rates are investigated, in detail, using a novel analysis based on the compressible enstrophy transport equation (CETE). Accurate analysis of the instability necessitates the generation of high fidelity numerical solutions, and this is achieved by employing optimized numerical methods that enable high accuracy direct numerical simulation of compressible flows. To study the effects of compressibility induced by rotation alone, a low free stream Mach number and two high rotation rates are considered, as compared to that reported in the literature. Results demonstrate single-sided vortex shedding, the presence of significant compressibility in the flow field confirmed by local Mach number, and temperature and density gradient fields with transient formation of supersonic pockets noted for the higher rotation speed cases. The temporal instability is studied by analyzing the relative contributions of different terms in the CETE to the growth of enstrophy. As per the authors' knowledge, this is the first such research effort demonstrating an application of the CETE for instabilities. Analysis shows that viscous diffusion is the dominant mechanism in creating the flow instability with a secondary role played by the baroclinic mechanism.

Maximum amplification of enstrophy in three-dimensional Navier–Stokes flows

This investigation concerns a systematic search for potentially singular behavior in 3D Navier-Stokes flows. Enstrophy serves as a convenient indicator of the regularity of solutions to the Navier Stokes system --- as long as this quantity remains finite, the solutions are guaranteed to be smooth and satisfy the equations in the classical (pointwise) sense. However, there are no estimates available with finite a priori bounds on the growth of enstrophy and hence the regularity problem for the 3D Navier-Stokes system remains open. In order to quantify the maximum possible growth of enstrophy, we consider a family of PDE optimization problems in which initial conditions with prescribed enstrophy $\mathcal{E}_0$ are sought such that the enstrophy in the resulting Navier-Stokes flow is maximized at some time $T$. Such problems are solved computationally using a large-scale adjoint-based gradient approach derived in the continuous setting. By solving these problems for a broad range of values of $\mathcal{E}_0$ and $T$, we demonstrate that the maximum growth of enstrophy is in fact finite and scales in proportion to $\mathcal{E}_0^{3/2}$ as $\mathcal{E}_0$ becomes large. Thus, in such worst-case scenario the enstrophy still remains bounded for all times and there is no evidence for formation of singularity in finite time. We also analyze properties of the Navier-Stokes flows leading to the extreme enstrophy values and show that this behavior is realized by a series of vortex reconnection events.

A Regularity Criterion for the Navier–Stokes Equation Involving Only the Middle Eigenvalue of the Strain Tensor

This manuscript derives an evolution equation for the symmetric part of the gradient of the velocity (the strain tensor) in the incompressible Navier-Stokes equation on $\mathbb{R}^3$, and proves the existence of $L^2$ mild solutions to this equation. We use this equation to obtain a simplified identity for the growth of enstrophy for mild solutions that depends only on the strain tensor, not on the nonlocal interaction of the strain tensor with the vorticity. The resulting identity allows us to prove a new family of scale-critical, necessary and sufficient conditions for the blow-up of a solution at some finite time $T_{max}<+\infty$, which depend only on the history of the positive part of the second eigenvalue of the strain matrix. Since this matrix is trace-free, this severely restricts the geometry of any finite-time blow-up. This regularity criterion provides analytic evidence of the numerically observed tendency of the vorticity to align with the eigenvector corresponding to the middle eigenvalue of the strain matrix. This regularity criterion also allows us to prove as a corollary a new scale critical, one component type, regularity criterion for a range of exponents for which there were previously no known critical, one component type regularity criteria. Furthermore, our analysis permits us to extend the known time of existence of smooth solutions with fixed initial enstrophy $E_0=\frac{1}{2}\left\|\nabla \otimes u^0\right \|_{L^2}^2$ by a factor of $4{,}920{.}75$---although the previous constant in the literature was not expected to be close to optimal, so this improvement is less drastic than it sounds, especially compared with numerical results. Finally, we will prove the existence and stability of blow-up for a toy model ODE for the strain equation.

Transient growth in stochastic Burgers flows

This study considers the problem of the extreme behavior exhibited by solutions to Burgers equation subject to stochastic forcing. More specifically, we are interested in the maximum growth achieved by the 'enstrophy' (the Sobolev $H^1$ seminorm of the solution) as a function of the initial enstrophy $\mathcal{E}_0$, in particular, whether in the stochastic setting this growth is different than in the deterministic case considered by Ayala &amp; Protas (2011). This problem is motivated by questions about the effect of noise on the possible singularity formation in hydrodynamic models. The main quantities of interest in the stochastic problem are the expected value of the enstrophy and the enstrophy of the expected value of the solution. The stochastic Burgers equation is solved numerically with a Monte Carlo sampling approach. By studying solutions obtained for a range of optimal initial data and different noise magnitudes, we reveal different solution behaviors and it is demonstrated that the two quantities always bracket the enstrophy of the deterministic solution. The key finding is that the expected values of the enstrophy exhibit the same power-law dependence on the initial enstrophy $\mathcal{E}_0$ as reported in the deterministic case. This indicates that the stochastic excitation does not increase the extreme enstrophy growth beyond what is already observed in the deterministic case.

Maximum Rate of Growth of Enstrophy in Solutions of the Fractional Burgers Equation

This investigation is a part of a research program aiming to characterize the extreme behavior possible in hydrodynamic models by analyzing the maximum growth of certain fundamental quantities. We consider here the rate of growth of the classical and fractional enstrophy in the fractional Burgers equation in the subcritical and supercritical regimes. Since solutions to this equation exhibit, respectively, globally well-posed behavior and finite-time blow-up in these two regimes, this makes it a useful model to study the maximum instantaneous growth of enstrophy possible in these two distinct situations. First, we obtain estimates on the rates of growth and then show that these estimates are sharp up to numerical prefactors. This is done by numerically solving suitably defined constrained maximization problems and then demonstrating that for different values of the fractional dissipation exponent the obtained maximizers saturate the upper bounds in the estimates as the enstrophy increases. We conclude that the power-law dependence of the enstrophy rate of growth on the fractional dissipation exponent has the same global form in the subcritical, critical and parts of the supercritical regime. This indicates that the maximum enstrophy rate of growth changes smoothly as global well-posedness is lost when the fractional dissipation exponent attains supercritical values. In addition, nontrivial behavior is revealed for the maximum rate of growth of the fractional enstrophy obtained for small values of the fractional dissipation exponents. We also characterize the structure of the maximizers in different cases.

Response of turbulent enstrophy to sudden implementation of spanwise wall oscillation in channel flow

The response of turbulent enstrophy to a sudden implementation of spanwise wall oscillation (SWO) is studied in a turbulent channel flow via direct numerical simulation. In the beginning of the application of SWO, a significant correlation is formed between ωy′ and ωz′. A transient growth of turbulent enstrophy occurs, which directly enhances turbulent dissipation and drifts the turbulent flow towards a new lower-drag condition. Afterwards, the terms related to the stretching of vorticity (ωx, ωy′, and ωz), the inclination of ωy′ by \(\frac{\partial w}{\partial y}\), the turn of ωz by \(\frac{\partial v\prime}{\partial z}\), and the horizontal shear of ωz by \(\frac{\partial w\prime}{\partial x}\) are suppressed due to the presence of SWO, leading to attenuation of the turbulent enstrophy.

Extreme vortex states and the growth of enstrophy in three-dimensional incompressible flows

In this investigation we study extreme vortex states defined as incompressible velocity fields with prescribed enstrophy${\mathcal{E}}_{0}$which maximize the instantaneous rate of growth of enstrophy$\text{d}{\mathcal{E}}/\text{d}t$. We provide an analytic characterization of these extreme vortex states in the limit of vanishing enstrophy${\mathcal{E}}_{0}$and, in particular, show that the Taylor–Green vortex is in fact a local maximizer of$\text{d}{\mathcal{E}}/\text{d}t$in this limit. For finite values of enstrophy, the extreme vortex states are computed numerically by solving a constrained variational optimization problem using a suitable gradient method. In combination with a continuation approach, this allows us to construct an entire family of maximizing vortex states parameterized by their enstrophy. We also confirm the findings of the seminal study by Lu &amp; Doering (Indiana Univ. Math. J., vol. 57, 2008, pp. 2693–2727) that these extreme vortex states saturate (up to a numerical prefactor) the fundamental bound$\text{d}{\mathcal{E}}/\text{d}t&lt;C{\mathcal{E}}^{3}$, for some constant$C&gt;0$. The time evolution corresponding to these extreme vortex states leads to a larger growth of enstrophy than the growth achieved by any of the commonly used initial conditions with the same enstrophy${\mathcal{E}}_{0}$. However, based on several different diagnostics, there is no evidence of any tendency towards singularity formation in finite time. Finally, we discuss possible physical reasons why the initially large growth of enstrophy is not sustained for longer times.

Turbulence and vorticity in Galaxy clusters generated by structure formation

Turbulence is a key ingredient for the evolution of the intracluster medium, whose properties can be predicted with high-resolution numerical simulations. We present initial results on the generation of solenoidal and compressive turbulence in the intracluster medium during the formation of a small-size cluster using highly resolved, non-radiative cosmological simulations, with a refined monitoring in time. In this first of a series of papers, we closely look at one simulated cluster whose formation was distinguished by a merger around z ∼ 0.3. We separate laminar gas motions, turbulence and shocks with dedicated filtering strategies and distinguish the solenoidal and compressive components of the gas flows using Hodge–Helmholtz decomposition. Solenoidal turbulence dominates the dissipation of turbulent motions (∼95 per cent) in the central cluster volume at all epochs. The dissipation via compressive modes is found to be more important (∼30 per cent of the total) only at large radii (≥0.5rvir) and close to merger events. We show that enstrophy (vorticity squared) is good proxy of solenoidal turbulence. All terms ruling the evolution of enstrophy (i.e. baroclinic, compressive, stretching and advective terms) are found to be significant, but in amounts that vary with time and location. Two important trends for the growth of enstrophy in our simulation are identified: first, enstrophy is continuously accreted into the cluster from the outside, and most of that accreted enstrophy is generated near the outer accretion shocks by baroclinic and compressive processes. Secondly, in the cluster interior vortex, stretching is dominant, although the other terms also contribute substantially.

Enstrophy and passive scalar transport near the turbulent/non-turbulent interface in a turbulent planar jet flow

The enstrophy (ω2/2) and passive scalar (ϕ) transport near the turbulent/non-turbulent (T/NT) interface is investigated using direct numerical simulation of a planar jet with passive scalar transport. To take into account the interface movement, we derive the transport equations for the enstrophy and the scalar in a local coordinate system moving with the T/NT interface. The characteristics of the T/NT interface are analyzed for three interface orientations. The cross-streamwise edge and the leading edge face the cross-streamwise and streamwise directions, respectively, and the trailing edge is opposite to the leading edge. The propagation velocity of the T/NT interface is derived from the enstrophy transport equation in the local coordinate system. The T/NT interface propagates toward the non-turbulent region on average at the cross-streamwise and leading edges, whereas the trailing edge frequently propagates into the turbulent region. The conditional average of the enstrophy transport equation in the local coordinate system shows that viscous diffusion transports, toward the non-turbulent region, enstrophy, that is advected from the turbulent core region or is produced slightly inside the T/NT interface. Viscous diffusion contributes greatly to the enstrophy growth in the region very close to the T/NT interface. The transport equation for the scalar ϕ in the local coordinate system is used to analyze the scalar transport near the T/NT interface. The conditional average of the advection term shows that ϕ in the non-turbulent region is frequently transported into the turbulent region across the cross-streamwise and leading edges by interface propagation toward the non-turbulent region. In contrast, ϕ in the turbulent region is frequently transported into the non-turbulent region across the trailing edge. The conditional averages of the advection and molecular diffusion terms show that both the interface propagation and the molecular diffusion contribute to the scalar transport across the T/NT interface.

Amplitude–phase decompositions and the growth and decay of solutions of the incompressible Navier–Stokes and Euler equations

We introduce the concept of amplitude–phase decompositions and apply it to the study of growth and decay of solutions of the incompressible Navier–Stokes and Euler equations. Amplitudes coincide with functionals of physical and mathematical interest. We are able to explicitly solve for the amplitudes in terms of the phases. The results obtained provide insights into the growth and decay of enstrophy and viscous dissipation, and identify new criteria for solutions to remain smooth for all time or blow-up in finite time.

Sharp bounds on enstrophy growth in the viscous Burgers equation

We use the Cole–Hopf transformation and the Laplace method for the heat equation to justify the numerical results on enstrophy growth in the viscous Burgers equation on the unit circle. We show that the maximum enstrophy achieved in the time evolution is scaled as , where is the large initial enstrophy, whereas the time needed for reaching the maximal enstrophy is scaled as . These bounds are sharp for initial conditions given by odd C 3 functions that are convex on half-period.

Numerical study on comparison of Navier-Stokes and Burgers equations

We compare freely decaying evolution of the Navier-Stokes equations with that of the 3D Burgers equations with the same kinematic viscosity and the same incompressible initial data by using direct numerical simulations. The Burgers equations are well-known to be regular by a maximum principle [A. A. Kiselev and O. A. Ladyzenskaya, “On existence and uniqueness of the solutions of the nonstationary problem for a viscous incompressible fluid,” Izv. Akad. Nauk SSSR Ser. Mat. 21, 655 (1957); A. A. Kiselev and O. A. Ladyzenskaya, Am. Math. Soc. Transl. 24, 79 (1957)] unlike the Navier-Stokes equations. It is found in the Burgers equations that the potential part of velocity becomes large in comparison with the solenoidal part which decays more quickly. The probability distribution of the nonlocal term \documentclass[12pt]{minimal}\begin{document}$-{\bm u}\cdot \nabla p$\end{document}−u·∇p, which spoils the maximum principle, in the local energy budget is studied in detail. It is basically symmetric, i.e., it can be either positive or negative with fluctuations. Its joint probability density functions with \documentclass[12pt]{minimal}\begin{document}$\frac{1}{2}|{\bm u}|^2$\end{document}12|u|2 and with \documentclass[12pt]{minimal}\begin{document}$\frac{1}{2}|{\bm \omega }|^2$\end{document}12|ω|2 are also found to be symmetric, fluctuating at the same times as the probability density function of \documentclass[12pt]{minimal}\begin{document}$-{\bm u}\cdot \nabla p$\end{document}−u·∇p. A power-law relationship is found in the mathematical bound for the enstrophy growth \documentclass[12pt]{minimal}\begin{document}$\dfrac{dQ}{dt} + 2 \nu P \propto \left(Q^a P^b\right)^\alpha ,$\end{document}dQdt+2νP∝QaPbα, where Q and P denote the enstrophy and the palinstrophy, respectively, and the exponents a and b are determined by calculus inequalities. We propose to quantify nonlinearity depletion by the exponent α on this basis.

Enstrophy growth in the viscous Burgers equation

We study bounds on the enstrophy growth for solutions of the viscous Burgers equation on the unit circle. Using the variational formulation of Lu and Doering, we prove that the maximizer of the enstrophy's rate of change is sharp in the limit of large enstrophy up to a numerical constant but does not saturate the Poincare inequality for mean-zero 1-periodic functions. Using the dynamical system methods, we give an asymptotic representation of the maximizer in the limit of large enstrophy as a viscous shock on the background of a linear rarefactive wave. This asymptotic construction is used to prove that a larger growth of enstrophy can be achieved when the initial data to the viscous Burgers equation saturates the Poincare inequality up to a numerical constant. An exact self-similar solution of the Burgers equation is constructed to describe formation of a metastable viscous shock on the background of a linear rarefactive wave. When we consider the Burgers equation on an infinite line subject to the nonzero (shock-type) boundary conditions, we prove that the maximum enstrophy achieved in the time evolution is scaled as E 3/2 , where E is the large initial enstrophy, whereas the time needed for reaching the maximal enstrophy is scaled as E −1/2 log(E). Similar but slower rates are proved on the unit circle.

On the Maximum Enstrophy Growth in Burgers Equation

The regularity of solutions of the three-dimensional Navier-Stokes equation is controlled by the boundedness of the enstrophy ε. The best estimate available to-date for its rate of growth is dε/dt ≤ Cε3, where C > 0, which was recently found to be sharp by Lu & Doering (2008). Applying straightforward time-integration to this instantaneous estimate leads to the possibility of loss of regularity in finite time, the so-called "blow-up", and therefore the central question is to establish sharpness of such finite-time bounds. We consider an analogous problem for Burgers equation which is used as a "toy model". The problem of saturation of finite-time estimates for the enstrophy growth is stated as a PDE-constrained optimization problemwhere the control variable ϕ represents the initial condition, which is solved numerically for a wide range of time windows T > 0 and initial enstrophies ε0. We find that the maximum enstrophy growth in finite time scales as ε0α with α ≈ 3/2. The exponent is smaller than α = 3 predicted by analytic means, therefore suggesting lack of sharpness of analytical estimates.

On maximum enstrophy growth in a hydrodynamic system

Enstrophy E plays an important role in the study of regularity of solutions to the three-dimensional Navier–Stokes equation. The best estimates for its growth available to-date do not rule out the possibility of enstrophy becoming unbounded in finite time which would indicate loss of regularity of solutions. It is therefore interesting to investigate sharpness of such finite-time bounds for enstrophy growth. We consider this question in the context of Burgers equation which is used as a “toy model”. The problem of saturation of finite-time estimates for the enstrophy growth is stated as a PDE-constrained optimization problem max ϕ [ E ( T ) − E ( 0 ) ] subject to E ( 0 ) = E 0 , where the control variable ϕ represents the initial condition, which is solved numerically using an adjoint-based gradient method for a wide range of time windows T and initial enstrophies E 0 . We show that this optimization problem admits a discrete family of maximizers parameterized by the wavenumber m whose members are rescaled copies of the fundamental maximizer corresponding to m = 1 . It is found that the maximum enstrophy growth in finite-time scales with the initial enstrophy as E 0 α where α ≈ 3 / 2 . The exponent is smaller than α = 3 predicted by analytic means, therefore suggesting possible lack of sharpness of analytical estimates.