Forced Burgers Equation Research Articles

On solutions to the Cauchy problem for a class of forced Burgers equations

Summary A Cauchy problem for forced Burgers equation (FBE) with time-dependent coefficients is studied in this work. We find explicit solutions of the FBE in terms of Hermite polynomials subject to various classes of initial data. Further, we prove the existence of solutions of FBE with respect to the bounded, continuous and summable initial data. We then derive an approximate solution to the Cauchy problem and find the error estimations between the true and approximate solutions. We give asymptotic profiles of the solutions depending on the initial data.

Forced Burgers equation with sticky impulsion source

Forced Burgers equation with sticky impulsion source

Fast reaction of soluble surfactant can remobilize a stagnant cap

Analytical solutions are derived showing that a stagnant cap of surfactant at the interface between two viscous fluids caused by a linear extensional flow can be remobilized by fast kinetic exchange of surfactant with one of the fluids. Using a complex variable formulation of this multiphysics problem at zero capillary number, zero Reynolds number and zero bulk Péclet number, and assuming a linear equation of state, it is shown that the system is governed by a forced complex Burgers equation at arbitrary surface Péclet number. Consequently, this nonlinear system is shown to be linearizable using a complex analogue of the Cole–Hopf transformation. Steady equilibria of the system at any finite value of the surface Péclet number are found explicitly in terms of parabolic cylinder functions. While surface diffusion is naturally expected to mollify sharp gradients associated with stagnant caps and to remobilize the interface, this work gives an analytical demonstration of the less intuitive result that fast kinetic exchange has a similar effect. Indeed, the analytical approach here imposes no limit on the surface Péclet number, which can be taken to be infinitely large so that surface diffusion is completely absent. Mathematically, the solution structure is then very rich allowing a theoretical investigation of this extreme case where it is seen that fast surfactant exchange with the bulk can alone remobilize a stagnant cap. Remarkably, it is also possible to track explicitly the time evolution of the system to these remobilized equilibria by finding time-evolving exact solutions.

Dynamic multiscaling in stochastically forced Burgers turbulence

We carry out a detailed study of dynamic multiscaling in the turbulent nonequilibrium, but statistically steady, state of the stochastically forced one-dimensional Burgers equation. We introduce the concept of interval collapse time, which we define as the time taken for a spatial interval, demarcated by a pair of Lagrangian tracers, to collapse at a shock. By calculating the dynamic scaling exponents of the moments of various orders of these interval collapse times, we show that (a) there is not one but an infinity of characteristic time scales and (b) the probability distribution function of the interval collapse times is non-Gaussian and has a power-law tail. Our study is based on (a) a theoretical framework that allows us to obtain dynamic-multiscaling exponents analytically, (b) extensive direct numerical simulations, and (c) a careful comparison of the results of (a) and (b). We discuss possible generalizations of our work to higher dimensions, for the stochastically forced Burgers equation, and to other compressible flows that exhibit turbulence with shocks.

How to Extract a Spectrum from Hydrodynamic Equations

The practical results gained from statistical theories of turbulence usually appear in the form of an inertial range energy spectrum $${\mathcal {E}}(k)\sim k^{-q}$$ and a cutoff wavenumber $$k_{c}$$ . For example, the values $$q=5/3$$ and $$\ell k_{c}\sim \mathrm {Re}^{3/4}$$ are intimately associated with Kolmogorov’s 1941 theory. To extract such spectral information from the Navier–Stokes equations, Doering and Gibbon (Phys. D 165, 163–175, 2020) introduced the idea of forming a set of dynamic wavenumbers $$\kappa _n(t)$$ from ratios of norms of solutions. The time averages of the $$\kappa _n(t)$$ can be interpreted as the 2nth moments of the energy spectrum. They found that $$1 < q \leqslant 8/3$$ , thereby confirming the earlier work of Sulem and Frisch (J. Fluid Mech. 72, 417–423, 1975) who showed that when spatial intermittency is included, no inertial range can exist in the limit of vanishing viscosity unless $$q \leqslant 8/3$$ . Since the $$\kappa _n(t)$$ are based on Navier–Stokes weak solutions, this approach connects empirical predictions of the energy spectrum with the mathematical analysis of the Navier–Stokes equations. This method is developed to show how it can be applied to many hydrodynamic models such as the two dimensional Navier–Stokes equations (in both the direct- and inverse-cascade regimes), the forced Burgers equation and shell models.

Study of the Bacterial "Conversations" and Pattern Formation in the Quorum Sensing System using Numerical Simulation

Cell-to-cell communication processes in the bacterial world can be considered as a collective bacterial behavior, which is coordinated by chemical signaling molecules (autoinducers, quorum sensing molecules, or pheromones). This complex biological process is termed the quorum sensing mechanism, which is considered a density-dependent bacterial communication system. As the bacterial culture grows, signal molecules are released into the extracellular milieu and accumulate, changing water fluidity. Under such threshold conditions, swimming bacterial suspensions impose a coordinated water movement on a length scale of the order of 10 to 100 micrometers compared with a bacterial size of the order of 3 micrometers. Here, we propose a non-local hydrodynamics of the quorum state and wave-like pattern formation using the forced Burgers equation with Kwak transformation. Such an approach resulted in the conversion of the Burgers equation paradigm into a reaction-diffusion system. The examination of the dynamics of the quorum sensing system, both analytically as well as numerically, results in similar long-time dynamical behaviour. Moreover, we find out the range kinematics viscosity of the living fluid, which is one of the significant parameters for pattern formation in the system.AMS Subject Classification: 92B05, 65N06, 65Z05. Doi: 10.28991/HEF-SP2022-01-03 Full Text: PDF

On end-point distribution for directed polymers and related problems for randomly forced Burgers equation.

In this paper, we study several problems related to the theory of randomly forced Burgers equation. Our numerical analysis indicates that despite the localization effects the quenched variance of the endpoint distribution for directed polymers in the strong disorder regime grows as the polymer length [Formula: see text]. We also present numerical results in support of the 'one force-one solution' principle. This article is part of the theme issue 'Scaling the turbulence edifice (part 2)'.

The Gaussian structure of the singular stochastic Burgers equation

AbstractWe consider the stochastically forced Burgers equation with an emphasis on spatially rough driving noise. We show that the law of the process at a fixed timet, conditioned on no explosions, is absolutely continuous with respect to the stochastic heat equation obtained by removing the nonlinearity from the equation. This establishes a form of ellipticity in this infinite-dimensional setting. The results follow from a recasting of the Girsanov Theorem to handle less spatially regular solutions while only proving absolute continuity at a fixed time and not on path-space. The results are proven by decomposing the solution into the sum of auxiliary processes, which are then shown to be absolutely continuous in law to a stochastic heat equation. The number of levels in this decomposition diverges to infinite as we move to the stochastically forced Burgers equation associated to the KPZ equation, which we conjecture is just beyond the validity of our results (and certainly the current proof). The analysis provides insights into the structure of the solution as we approach the regularity of KPZ. A number of techniques from singular SPDEs are employed, as we are beyond the regime of classical solutions for much of the paper.

Subgrid-scale parametrization of unresolved scales in forced Burgers equation using generative adversarial networks (GAN)

Stochastic subgrid-scale parametrizations aim to incorporate effects of unresolved processes in an effective model by sampling from a distribution usually described in terms of resolved modes. This is an active research area in climate, weather and ocean science where processes evolved in a wide range of spatial and temporal scales. In this study, we evaluate the performance of conditional generative adversarial network (GAN) in parametrizing subgrid-scale effects in a finite-difference discretization of stochastically forced Burgers equation. We define resolved modes as local spatial averages and deviations from these averages are the unresolved degrees of freedom. We train a Wasserstein GAN (WGAN) conditioned on the resolved variables to learn the distribution of subgrid flux tendencies for resolved modes and, thus, represent the effect of unresolved scales. The resulting WGAN is then used in an effective model to reproduce the statistical features of resolved modes. We demonstrate that various stationary statistical quantities such as spectrum, moments, autocorrelation, etc. are well approximated by this effective model.

A new type of Darboux transformations for the one-dimensional Burgers equation with forcing

We construct new types of arbitrary-order Darboux transformations for the stationary forced Burgers equation in one dimension. The distinguishing property of these transformations is that they generate a forcing term in the transformed Burgers equation that is inherently a one-parameter family of functions with linear dependence on the parameter. Our construction is based on an adaptation of results for Schrödinger-type equations [].

Exact solutions for the formation of stagnant caps of insoluble surfactant on a planar free surface

A class of exact solutions is presented describing the time evolution of insoluble surfactant to a stagnant cap equilibrium on the surface of deep water in the Stokes flow regime at zero capillary number and infinite surface Péclet number. This is done by demonstrating, in a two-dimensional model setting, the relevance of the forced complex Burgers equation to this problem when a linear equation of state relates the surface tension to the surfactant concentration. A complex-variable version of the method of characteristics can then be deployed to find an implicit representation of the general solution. A special class of initial conditions is considered for which the associated solutions can be given explicitly. The new exact solutions, which include both spreading and compactifying scenarios, provide analytical insight into the unsteady formation of stagnant caps of insoluble surfactant. It is also shown that first-order reaction kinetics modelling sublimation or evaporation of the insoluble surfactant to the upper gas phase can be incorporated into the framework; this leads to a forced complex Burgers equation with linear damping. Generalized exact solutions to the latter equation at infinite surface Péclet number are also found and used to study how reaction effects destroy the surfactant cap equilibrium.

Dynamical properties of generalized traveling waves of exactly solvable forced Burgers equations with variable coefficients

The initial value problem for a generalized forced Burgers equation with variable coefficients Ut+(μ˙(t)/μ(t))U+UUx=(1/2μ(t))Uxx−a(t)Ux+b(t)(xU)x−ω2(t)x+f(t),x∈R,t>0, is solved using Cole-Hopf linearization and Wei-Norman Lie algebraic approach for finding the evolution operator of the associated linear diffusion type equation. As a result, solution of the initial value problem is obtained in terms of a corresponding linear second-order inhomogeneous ordinary differential equation and a standard Burgers model. Then, using the translation and Galilean invariance of standard Burgers equation, families of generalized nonlinear waves propagating according to a Newtonian type equation of motion are constructed. The influence of the damping, dilatation and forcing terms on the dynamics of shocks, multi-shocks, triangular and N-shaped generalized traveling waves and rational type solutions with moving singularities is investigated. Finally, exactly solvable models with concrete time-variable coefficients are introduced and dynamical properties of certain particular solutions are discussed.

Dirichlet problem on the half-line for a forced Burgers equation with time-variable coefficients and exactly solvable models

We consider a forced Burgers equation with time-variable coefficients and solve the initial-boundary value problem on the half-line 0 < x < ∞ with inhomogeneous Dirichlet boundary condition imposed at x=0. Solution of this problem is obtained in terms of a corresponding second order ordinary differential equation and a second kind singular Volterra type integral equation. As an application of the general results, we introduce three different Burgers type models with specific damping, diffusion and forcing coefficients and construct classes of exactly solvable models. The Burgers problems with smooth time-dependent boundary data and an initial profile with pole type singularity have exact solutions with moving singularity. For each model we provide the solutions explicitly and describe the dynamical properties of the singularities depending on the time-variable coefficients and the given initial and boundary data.

On non-autonomously forced Burgers equation with periodic and Dirichlet boundary conditions

AbstractWe study the non-autonomously forced Burgers equation $$u_t(x,t) + u(x,t)u_x(x,t)-u_{xx}(x,t) = f(x,t)$$ on the space interval (0, 1) with two sets of the boundary conditions: the Dirichlet and periodic ones. For both situations we prove that there exists the unique H1 bounded trajectory of this equation defined for all t ∈ ℝ. Moreover we demonstrate that this trajectory attracts all trajectories both in pullback and forward sense. We also prove that for the Dirichlet case this attraction is exponential.

Green’s functions and the Cauchy problem of the Burgers hierarchy and forced Burgers equation

We consider the Cauchy problem for the Burgers hierarchy with general time dependent coefficients. The closed form for the Green’s function of the corresponding linear equation of arbitrary order N is shown to be a sum of generalised hypergeometric functions. For suitably damped initial conditions we plot the time dependence of the Cauchy problem over a range of N values. For N=1, we introduce a spatial forcing term. Using connections between the associated second order linear Schrödinger and Fokker-Planck equations, we give closed form expressions for the corresponding Green’s functions of the sinked Bessel process with constant drift. We then apply the Green’s function to give time dependent profiles for the corresponding forced Burgers Cauchy problem.

Energy flux enhancement, intermittency and turbulence via Fourier triad phase dynamics in the 1-D Burgers equation

We present a theoretical and numerical study of Fourier-space triad phase dynamics in the one-dimensional stochastically forced Burgers equation at Reynolds number $Re\approx 2.7\times 10^{4}$. We demonstrate that Fourier triad phases over the inertial range display a collective behaviour characterised by intermittent periods of synchronisation and alignment, reminiscent of the Kuramoto model (Chemical Oscillations, Waves, and Turbulence, Springer, 1984) and directly related to collisions of shocks in physical space. These periods of synchronisation favour efficient energy fluxes across the inertial range towards small scales, resulting in strong bursts of dissipation and enhanced coherence of the Fourier energy spectrum. The fast time scale of the onset of synchronisation relegates energy dynamics to a passive role: this is further examined using a reduced system with the Fourier amplitudes fixed in time – a phase-only model. We show that intermittent triad phase dynamics persists without amplitude evolution and we broadly recover many of the characteristics of the full Burgers system. In addition, for both full Burgers and phase-only systems the physical-space velocity statistics reveals that triad phase alignment is directly related to the non-Gaussian statistics typically associated with structure-function intermittency in turbulent systems.

Variational multiscale approximation of the one‐dimensional forced Burgers equation: The role of orthogonal subgrid scales in turbulence modeling

SummaryA numerical approximation for the one‐dimensional Burgers equation is proposed by means of the orthogonal subgrid scales–variational multiscale (OSGS‐VMS) method. We evaluate the role of the variational subscales in describing the Burgers “turbulence” phenomena. Particularly, we seek to clarify the interaction between the subscales and the resolved scales when the former are defined to be orthogonal to the finite‐dimensional space. Direct numerical simulation is used to evaluate the resulting OSGS‐VMS energy spectra. The comparison against a large eddy simulation model is presented for numerical discretizations in which the grid is not capable of solving the small scales. An accurate approximation to the phenomena of turbulence is obtained with the addition of the purely dissipative numerical terms given by the OSGS‐VMS method without any modification of the continuous problem.

Large Time Asymptotics with Error Estimates to Solutions of a Forced Burgers Equation

This article deals with a forced Burgers equation (FBE) subject to the initial function, which is continuous and summable on . Large time asymptotic behavior of solutions to the FBE is determined with precise error estimates. To achieve this, we construct solutions for the FBE with a different initial class of functions using the method of separation of variables and Cole–Hopf like transformation. These solutions are constructed in terms of Hermite polynomials with the help of similarity variables. The constructed solutions would help us to pick up an asymptotic approximation and to show that the magnitude of the difference function of the true and approximate solutions decays algebraically to 0 for large time.

Intermittency in fractal Fourier hydrodynamics: Lessons from the Burgers equation.

We present theoretical and numerical results for the one-dimensional stochastically forced Burgers equation decimated on a fractal Fourier set of dimension D. We investigate the robustness of the energy transfer mechanism and of the small-scale statistical fluctuations by changing D. We find that a very small percentage of mode-reduction (D ≲ 1) is enough to destroy most of the characteristics of the original nondecimated equation. In particular, we observe a suppression of intermittent fluctuations for D < 1 and a quasisingular transition from the fully intermittent (D=1) to the nonintermittent case for D ≲ 1. Our results indicate that the existence of strong localized structures (shocks) in the one-dimensional Burgers equation is the result of highly entangled correlations amongst all Fourier modes.

Phase and precession evolution in the Burgers equation

We present a phenomenological study of the phase dynamics of the one-dimensional stochastically forced Burgers equation, and of the same equation under a Fourier mode reduction on a fractal set. We study the connection between coherent structures in real space and the evolution of triads in Fourier space. Concerning the one-dimensional case, we find that triad phases show alignments and synchronisations that favour energy fluxes towards small scales --a direct cascade. In addition, strongly dissipative real-space structures are associated with entangled correlations amongst the phase precession frequencies and the amplitude evolution of Fourier triads. As a result, triad precession frequencies show a non-Gaussian distribution with multiple peaks and fat tails, and there is a significant correlation between triad precession frequencies and amplitude growth. Links with dynamical systems approach are briefly discussed, such as the role of unstable critical points in state space. On the other hand, by reducing the fractal dimension D of the underlying Fourier set, we observe: i) a tendency toward a more Gaussian statistics, ii) a loss of alignment of triad phases leading to a depletion of the energy flux, and iii) the simultaneous reduction of the correlation between the growth of Fourier mode amplitudes and the precession frequencies of triad phases.