Incompressible Velocity Field Research Articles

Identifying the body force from partial observations of a two-dimensional incompressible velocity field

Identifying the body force from partial observations of a two-dimensional incompressible velocity field

Mathematical modeling of particle movement in laminar flow in a pipe

The article provides basic information about the principles of calculations for solving the system of Navier–Stokes equations using the control volume method. The calculation of the incompressible fluid velocity field and pressure is found using the SIMPLE algorithm and MacCormack method. Particle trajectories were determined using the Lagrangian approach.

Conformal invariance of 2D quantum turbulence in an exciton–polariton fluid of light

The similarities of quantum turbulence with classical hydrodynamics allow quantum fluids to provide essential models of their classical analog, paving the way for fundamental advances in physics and technology. Recently, experiments on 2D quantum turbulence observed the clustering of same-sign vortices in strong analogy with the inverse energy cascade of classical fluids. However, self-similarity of the turbulent flow, a fundamental concept in the study of classical turbulence, has so far remained largely unexplored in quantum systems. Here, thanks to the unique features of exciton–polaritons, we measure the scale invariance of velocity circulations and show that the cascade process follows the universal scaling of critical phenomena in 2D. We demonstrate this behavior from the statistical analysis of the experimentally measured incompressible velocity field and the microscopic imaging of the quantum fluid. These results can find wide application in both quantum and classical 2D turbulence.

Global well-posedness and convergence to equilibrium for the Abels-Garcke-Grün model with nonlocal free energy

We investigate the nonlocal version of the Abels-Garcke-Grün (AGG) system, which describes the motion of a mixture of two viscous incompressible fluids. This consists of the incompressible Navier-Stokes-Cahn-Hilliard system characterized by concentration-dependent density and viscosity, and an additional flux term due to interface diffusion. In particular, the Cahn-Hilliard dynamics of the concentration (phase-field) is governed by the aggregation/diffusion competition of the nonlocal Helmholtz free energy with singular (logarithmic) potential and constant mobility. We first prove the existence of global strong solutions in general two-dimensional bounded domains and their uniqueness when the initial datum is strictly separated from the pure phases. The key points are a novel well-posedness result of strong solutions to the nonlocal convective Cahn-Hilliard equation with singular potential and constant mobility under minimal integral assumption on the incompressible velocity field, and a new two-dimensional interpolation estimate for the L4(Ω) control of the pressure in the stationary Stokes problem. Secondly, we show that any weak solution, whose existence was already known, is globally defined, enjoys the propagation of regularity and converges towards an equilibrium (i.e., a stationary solution) as t→∞. Furthermore, we demonstrate the uniqueness of strong solutions and their continuous dependence with respect to general (not necessarily separated) initial data in the case of matched densities and unmatched viscosities (i.e., the nonlocal model H with variable viscosity, singular potential and constant mobility). Finally, we provide a stability estimate between the strong solutions to the nonlocal AGG model and the nonlocal Model H in terms of the difference of densities.

Exponential mixing for random dynamical systems and an example of Pierrehumbert

We consider the question of exponential mixing for random dynamical systems on arbitrary compact manifolds without boundary. We put forward a robust, dynamics-based framework that allows us to construct space-time smooth, uniformly bounded in time, universal exponential mixers. The framework is then applied to the problem of proving exponential mixing in a classical example proposed by Pierrehumbert in 1994, consisting of alternating periodic shear flows with randomized phases. This settles a longstanding open problem on proving the existence of a space-time smooth (universal) exponentially mixing incompressible velocity field on a two-dimensional periodic domain while also providing a toolbox for constructing such smooth universal mixers in all dimensions.

Incompressible limit for a fluid mixture

In this paper we discuss the incompressible limit for multicomponent fluids in the isothermal ideal case. Both a direct limit-passage in the equation of state and the low Mach-number limit in rescaled PDEs are investigated. Using the relative energy inequality, we obtain convergence results for the densities and the velocity-field under the condition that the incompressible model possesses a sufficiently smooth solution, which is granted at least for a short time. Moreover, in comparison to single-component flows, uniform estimates and the convergence of the pressure are needed in the multicomponent case because the incompressible velocity field is not divergence-free. We show that certain constellations of the mobility tensor allow to control gradients of the entropic variables and yield the convergence of the pressure in L1.

Curl-Flow

We propose to augment standard grid-based fluid solvers with pointwise divergence-free velocity interpolation, thereby ensuring exact incompressibility down to the sub-cell level. Our method takes as input a discretely divergence-free velocity field generated by a staggered grid pressure projection, and first recovers a corresponding discrete vector potential. Instead of solving a costly vector Poisson problem for the potential, we develop a fast parallel sweeping strategy to find a candidate potential and apply a gauge transformation to enforce the Coulomb gauge condition and thereby make it numerically smooth. Interpolating this discrete potential generates a point-wise vector potential whose analytical curl is a pointwise incompressible velocity field. Our method further supports irregular solid geometry through the use of level set-based cut-cells and a novel Curl-Noise-inspired potential ramping procedure that simultaneously offers strictly non-penetrating velocities and incompressibility. Experimental comparisons demonstrate that the vector potential reconstruction procedure at the heart of our approach is consistently faster than prior such reconstruction schemes, especially those that solve vector Poisson problems. Moreover, in exchange for its modest extra cost, our overall Curl-Flow framework produces significantly improved particle trajectories that closely respect irregular obstacles, do not suffer from spurious sources or sinks, and yield superior particle distributions over time.

Incompressible energy spectrum from wave turbulence

Bose–Einstein condensates with their superfluidity property provide an interesting parallel to classical fluids. Due to the Kolmogorov spectrum of homogeneous turbulence the statistics of the incompressible velocity field is of great interest, but in superfluids obtaining quantities such as the statistics of the velocity field from the macroscopic wavefunction turns out be a complicated task; therefore, most of the work up to now has been numerical in nature. We made use of the Weak Wave Turbulence (WWT) theory, which provides the statistics of the macroscopic wavefunction, to obtain the statistics of the velocity field, which allowed us to produce a semi analytical procedure for extracting the incompressible energy spectrum in the WWT regime. This is done by introducing an auxiliary wavefunction that preserves the relevant statistical and hydrodynamical properties of the condensate but with a homogeneous density thus allowing for a simpler description of the velocity field.

Bilinear Control of Convection-Cooling: From Open-Loop to Closed-Loop

This paper is concerned with a bilinear control problem for enhancing convection-cooling via an incompressible velocity field. Both optimal open-loop control and closed-loop feedback control designs are addressed. First and second order optimality conditions for characterizing the optimal solution are discussed. In particular, the method of instantaneous control is applied to establish the feedback laws. Moreover, the construction of feedback laws is also investigated by directly utilizing the optimality system with appropriate numerical discretization schemes. Computationally, it is much easier to implement the closed-loop feedback control than the optimal open-loop control, as the latter requires to solve the state equations forward in time, coupled with the adjoint equations backward in time together with a nonlinear optimality condition. Rigorous analysis and numerical experiments are presented to demonstrate our ideas and validate the efficacy of the control designs.

A Pointwise Conservative Method for Thermochemical Convection Under the Compressible Anelastic Liquid Approximation

AbstractIn prior work we found that precise approximation of the continuity constraint is crucial for accurate propagation of tracer data when advected through a background incompressible velocity field (Sime et al., 2021, https://doi.org/10.1029/2020gc009349). Here we extend this investigation to compressible flows using the anelastic liquid approximation (ALA) and address four related issues: (a) Exact conservation of tracer discretized fields through a background compressible velocity; (b) Exact mass conservation; (c) Addition and removal of tracers without affecting (exact) conservation to preserve a consistent number of tracers per cell; and (d) the diffusion of tracer data, for example, as induced by thermal or chemical effects. In this process we also present an abstract formulation of the interior penalty hybrid discontinuous Galerkin (HDG) finite element formulation for diffusion problems and apply it to the advection‐diffusion and compressible Stokes systems. Finally we present numerical experiments exhibiting the HDG compressible Stokes momentum formulation's superconvergent compressibility approximation and reproduce examples of a community benchmark for the ALA.

Discrete maximum principle of a high order finite difference scheme for a generalized Allen–Cahn equation

We consider solving a generalized Allen-Cahn equation coupled with a passive convection for a given incompressible velocity field. The numerical scheme consists of the first order accurate stabilized implicit explicit time discretization and a fourth order accurate finite difference scheme, which is obtained from the finite difference formulation of the $Q^2$ spectral element method. We prove that the discrete maximum principle holds under suitable mesh size and time step constraints. The same result also applies to construct a bound-preserving scheme for any passive convection with an incompressible velocity field.

Initial–boundary value problems for merely bounded nearly incompressible vector fields in one space dimension

We establish existence and uniqueness results for initial-boundary value problems for transport equations in one space dimension with nearly incompressible velocity fields, under the sole assumption that the fields are bounded. In the case where the velocity field is either nonnegative or nonpositive, one can rely on similar techniques as in the case of the Cauchy problem. Conversely, in the general case we introduce a new and more technically demanding construction, which heuristically speaking relies on a “lagrangian formulation” of the problem, albeit in a highly irregular setting. We also establish stability of the solution in weak and strong topologies, and propagation of the BV regularity. In the case of either nonnegative or nonpositive velocity fields we also establish a BV-in-time regularity result, and we exhibit a counterexample showing that the result is false in the case of sign-changing vector fields. To conclude, we establish a trace renormalization property.

A fractional PDE model for turbulent velocity fields near solid walls

Abstract

Optimal control of convection-cooling and numerical implementation

This paper is concerned with the problem of enhancing convection-cooling via active control of the incompressible velocity field, described by a stationary diffusion–convection model. This essentially leads to a bilinear optimal control problem. A rigorous proof of the existence of an optimal control is presented and the first order optimality conditions are derived for solving the control using a variational inequality. Moreover, the second order sufficient conditions are established to characterize the local minimizer. Finally, numerical experiments are conducted utilizing finite elements methods together with nonlinear iterative schemes, to demonstrate and validate the effectiveness of our control design.

Generation of a random anisotropic incompressible turbulent velocity field in accordance with the calculated flow statistics

On the base of Randomized Spectral Method (RSM), a new stochastic algorithm for the generation of homogeneous anisotropic turbulent velocity fields has been developed. This technique provides incompressibility of the obtained velocity field realizations simultaneously with the two main required statistical properties—the Reynolds-stress tensor and the turbulent energy spectrum. The method is computationally efficient and can easily be extended to the case of inhomogeneous turbulence.

Tracer Turbulence: The Batchelor–Howells–Townsend Spectrum Revisited

Given a velocity field u(x, t), we consider the evolution of a passive tracer $$\theta $$ governed by $$\partial _t\theta + u\cdot \nabla \theta = \Delta \theta + g$$ with time-independent source g(x). When $$\Vert u\Vert $$ is small in some sense, Batchelor, Howells and Townsend (J Fluid Mech 5:134, 1959) predicted that the tracer spectrum scales as $$|\theta _k|^2\propto |k|^{-4}|u_k|^2$$. In this paper we prove that, for random synthetic two-dimensional incompressible velocity fields u(x, t) with given energy spectra, this scaling does indeed hold probabilistically, asymptotically almost surely for large |k| and small $$\Vert u\Vert $$. We also propose an asymptotic correction factor to the BHT scaling arising from the time-dependence of u.

Hamiltonian gyrofluid reductions of gyrokinetic equations

We provide a general framework for deriving Hamiltonian electromagnetic gyrofluid models from a Hamiltonian system of gyrokinetic equations. The presented procedure permits to derive gyrofluid models for an arbitrary number of moments with respect to the velocity coordinate parallel to an equilibrium magnetic field. The resulting gyrofluid models account, in particular, for finite Larmor radius effects, equilibrium temperature anisotropies and fluctuations of the magnetic field in the direction parallel to the equilibrium magnetic field, thus generalizing Hamiltonian gyrofluid models previously presented in the literature. The Hamiltonian reduction procedure leading from the parent gyrokinetic model to the gyrofluid models is formulated in two stages. In the first step, after having shown that the parent gyrokinetic system indeed posseses a Hamiltonian structure, a Hamiltonian system is derived, by means of a Poisson sub-algebra argument, which describes the evolution of the perturbation of the gyrocenter distribution function, averaged with respect to the magnetic moment coordinate. The second stage brings from the latter model to the gyrofluid models by means of a closure relation, applicable at an arbitrary order in the moment hierarchy, which guarantees the preservation of a Hamiltonian structure. Casimir invariants of the noncanonical Poisson brackets of the gyrofluid models are provided. It is also shown how, in the two-dimensional limit, the gyrofluid model equations can be cast in the form of advection equations for Lagrangian invariants transported by generalized incompressible velocity fields, thus extending results obtained for previous Hamiltonian gyrofluid and drift-fluid models. The Hamiltonian reduction procedure is applied to derive a five-field gyrofluid model evolving the first two moments for the electron species and the first three moments for the ion species. The Casimir invariants and the Lagrangian advection formulation are provided explicitly for the five-field model. Remarks concerning possible variants of the procedure are discussed. As an example, it is shown how, by means of a variant of the procedure, it is possible to derive an isothermal two-field model for kinetic Alfvén waves including equilibrium electron temperature anisotropy effects.

A relation between membrane permeability and flow rate at low Reynolds number in circular pipe

A relation between membrane permeability to a pure solvent and flow rate is derived in a circular pipe driven by a constant pressure difference across the pipe length. Membrane is assumed to be non-deformable and zero thickness, and the permeate flux of pure solvent due to pressure discontinuity across the flat membrane is coupled with the governing equations for incompressible Newtonian fluid in the Stokes regime. The permeation flow rate (normalised by that of the Hagen-Poiseuille flow for the no-membrane case or with a membrane of infinite permeability) is represented as a function of a non-dimensional permeability including the aspect ratio of the pipe geometry. The relation is established through comparison with a fully-validated numerical simulation result: the numerical discretisation is based on our original discrete-forcing immersed boundary method, which guarantees (i) conservations of mass and momentum even in the immediate vicinity of the membrane surface and (ii) consistency between incompressible velocity and pressure fields. Inverse analysis of the above formula yields the permeability as function of the flow rate through the pipe, comprising three equations covering the entire permeability ranges. The established permeability formulae are expected to be useful for identifying effective permeabilities of membrane to single-component fluid or pure solvent in practical applications.

Polynomial mixing under a certain stationary Euler flow

We study the mixing properties of a scalar ρ on the unit disk advected by a certain incompressible velocity field u, which is a stationary radial solution of the Euler equation. The scalar ρ solves the continuity equation with the velocity field u and we can measure the degree of “mixedness” of ρ with two different scales commonly used in this setting, namely the geometric and the functional mixing scale. We develop a physical space approach well adapted to the quantitative analysis of the decay in time of the geometric mixing scale, which turns out to be polynomial for a large class of initial data. This extends previous results for the functional mixing scale, based on the explicit expression for the solution in Fourier variable, results that are also partially recovered by our approach.

One‐dimensional turbulence investigation of incompressible and low Mach number variable density pipe‐flow

AbstractCylindrical pipe‐flow is evaluated by means of the One‐Dimensional Turbulence (ODT) model applying both an incompressible and a low Mach number variable density framework. In the variable density formulation, the velocity field is decomposed into density‐related (non‐divergence‐free) and density‐unrelated (divergence‐free) contributions, as detailed in Medina et al. [Combust. Flame 190 (2018) 388‐401]. Both contributions to the velocity field are subject to the effects of diffusion and turbulent transport, as it is normally the case for the incompressible velocity field in traditional ODT formulations. The turbulent transport is implemented by means of stochastic eddy events. These are influenced by the available kinetic energy given by the divergence‐free velocity contribution. An eddy may arise from the different regions of shear caused by varying velocity gradients in the 1‐D domain, but also as a consequence of the temperature‐varying dynamic viscosity. The cylindrical low Mach number variable density formulation is tested in a very low heat transfer regime and compared to the passive scalar results of the incompressible pipe‐flow formulation. ODT model parameters are assumed equal for both formulations. ODT results are compared to DNS results from Satake and Kunugi [Int. J. Numer. Methods Heat Fluid Flow 12 (2002) 958‐1008]. The passive scalar and variable density results show very good agreement with each other as well as with DNS data, respectively, thus suggesting new possible applications for ODT.