Divergence-free Velocity Field Research Articles

A principle of maximum entropy for the Navier–Stokes equations

A principle of maximum entropy is proposed in the context of viscous incompressible flow in Eulerian coordinates. The relative entropy functional, defined over the space of L2 divergence-free velocity fields, is maximized relative to alternate measures supported over the energy–enstrophy surface. Since thermodynamic equilibrium distributions are characterized by maximum entropy, connections are drawn with stationary statistical solutions of the incompressible Navier–Stokes equations. Special emphasis is on the correspondence with the final statistics described by Kolmogorov’s theory of fully developed turbulence.

Entropy damping and Bulk Viscosity based artificial compressibility methods on dynamically distorting grids

Artificial Compressibility Methods (ACM) rely on an artificial equation that links the pressure and velocity fields to model incompressible flows. These hyperbolic/parabolic equations can rapidly converge to a ‘nearly’ divergence-free flow field in contrast to the methods based on the elliptic pressure Poisson equation. We compare the computational efficacy of two ACMs, namely, the Bulk Viscosity ACM (BVACM) and Entropically Damped Artificial Compressibility (EDAC) recently proposed in the literature. The methods implemented in the in-house high-order finite difference solver, COMPSQUARE, are validated for the test cases of a 2D doubly periodic shear layer (DPSL), a 3D Taylor Green Vortex (TGV), and 2D/3D NACA0012 airfoil pitching about the quarter chord. The efficacy of these methods was also tested on static and dynamic grids using conservative metrics. Although both ACMs yield competitive results, the divergence of the velocity field is found to be more prominent in the highly unsteady regions. BVACM resulted in (a) a superior divergence-free velocity field and (b) 20−38% higher maximum stable time than the EDAC, thereby increasing the computational speed. A higher value of the bulk viscosity coefficient, A, although ensures a stringent divergence-free velocity field, is shown to have minimal effect on the flow statistics and reduce the maximum stable time step. The parabolic–hyperbolic nature of the governing equations and the lack of dual time-stepping in BVACM and EDAC ensures that both these methods are highly scalable on massively parallel architectures. Since the energy equation is no longer required to compute the velocity field, both EDAC and BVACM approaches are found to be 8−10% faster than the weakly compressible Navier–Stokes simulations under the low-Mach number limit.

A Fourier spectral immersed boundary method with exact translation invariance, improved boundary resolution, and a divergence-free velocity field

This paper introduces a new immersed boundary (IB) method for viscous incompressible flow, based on a Fourier spectral method for the fluid solver and on the nonuniform fast Fourier transform (NUFFT) algorithm for coupling the fluid with the immersed boundary. The new Fourier spectral immersed boundary (FSIB) method gives improved boundary resolution in comparison to the standard IB method. The interpolated velocity field, in which the boundary moves, is analytically divergence-free. The FSIB method is gridless and has the meritorious properties of volume conservation, exact translation invariance, conservation of momentum, and conservation of energy. We verify these advantages of the FSIB method numerically both for the Stokes equations and for the Navier-Stokes equations in both two and three space dimensions. The FSIB method converges faster than the IB method. In particular, we observe second-order convergence in various problems for the Navier-Stokes equations in three dimensions. The FSIB method is also computationally efficient with complexity of O(N3log⁡(N)) per time step for N3 Fourier modes in three dimensions.

An improved Coupled Level Set and Volume of Fluid (i-CLSVoF) framework for sessile droplet evaporation

Surface-tension-dominant droplet evaporation is ubiquitous and of importance to many applications. We present an improved Coupled Level Set and Volume of Fluid (i-CLSVoF) framework without explicit interface reconstruction for modelling micro-sized droplets with and without evaporation. In the i-CLSVoF framework, an improved surface tension force model with additional filtering steps to filter un-physical spurious velocities is developed and implemented. A simple, yet efficient, velocity-potential based approach is proposed to reconstruct a divergence-free velocity field for the advection of the free surface during droplet evaporation. This approach fixes the numerical issues resulting from the evaporation-induced velocity jump at the interface. The smeared mass source term approach incorporated in this work guarantees greater numerical stability than the non-smeared approach. Three different evaporation models (constant mass flux, thermally driven evaporation and droplet evaporation at room temperature) are implemented in the i-CLSVoF. Corresponding numerical benchmark cases (dam break, droplet relaxation and droplet evaporation subjected to different evaporation models) are conducted to validate the surface tension and the evaporation models. Good agreement between the numerical and corresponding analytical solutions is found. The model developed in this work shows convincing performance in modelling surface-tension-dominant flow with and without evaporation.

Stokesian processes : inferring Stokes flows using physics-informed Gaussian processes

We develop a probabilistic Stokes flow framework, using physics informed Gaussian processes, which can be used to solve both forward/inverse flow problems with missing and/or noisy data. The physics of the problem, specified by the Stokes and continuity equations, is exactly encoded into the inference framework. Crucially, this means that we do not need to explicitly solve the Poisson equation for the pressure field, as a physically meaningful (divergence-free) velocity field will automatically be selected. We test our method on a simple pressure driven flow problem, i.e. flow through a sinusoidal channel, and compare against standard numerical methods (Finite Element and Direct Numerical Simulations). We obtain excellent agreement, even when solving inverse problems given only sub-sampled velocity data on low dimensional sub-spaces (i.e. 1 component of the velocity on 1D domains to reconstruct 2D flows). The proposed method will be a valuable tool for analyzing experimental data, where noisy/missing data is the norm.

A new divergence-free synthetic eddy method for generating homogeneous isotropic turbulence with a prescribed energy spectrum

A new divergence-free synthetic eddy method (SEM) for the generation of homogeneous isotropic turbulence (HIT) with a prescribed energy spectrum is presented. In the proposed method, the divergence-free consistent synthetic velocity field is generated from the curl of a vector potential field following conventional SEM methodology. A carefully designed formula is constructed for the computation of the local vector potential field induced by each synthetic eddy, with which a desired velocity spectrum tensor can be satisfied for the synthetic incompressible turbulence exactly. The proposed method is characterized by a multi-scale feature in which the length scales of synthetic eddies are determined by a dimensionless random variable satisfying a particular probability distribution function (PDF). The mathematical relation, concerning the energy spectrum of the synthetic turbulence, the shape function of the eddies and the PDF of the eddy-scale-related random variable, is derived. This assists in determining either the shape function or the PDF. The performance of the proposed method is examined by synthesizing HIT satisfying a theoretical energy spectrum model and HIT identified in a wind-tunnel experiment.

On the conservative phase-field method with the N-component incompressible flows

This paper presents a conservative Allen–Cahn model coupled with the incompressible Navier–Stokes equation for tracking the interface with the N-component immiscible fluids system. The proposed conservative phase-field model can track the interface with large deformation in divergence-free velocity fields. The erroneous estimation of the normal vector is a challenging numerical issue for the interface capturing due to the appearance of spurious oscillations. The improved phase-field-based method combines the nonlinear preprocessing operation guided by the level-set method with local artificial viscosity stabilization to improve the computation of the discrete normal vector. The interfaces between different immiscible components are replaced by the transition region with finite thickness in the continuous phase field. The surface tension effects are represented with the continuous surface tension force in the system, which is not limited by the number of components. The third-order Runge-Kutta time discretization and second-order spatial discretization are applied for the multi-component system. To eliminate the spurious oscillations caused by discontinuous and steep gradient for capturing the shocks and sharp interfaces, we apply the third-order weighted essentially non-oscillatory method for the advection term. Several quantitative results of numerical tests, such as error estimation with the proposed method, comparative tests with different methods, and convergence rate for classical benchmark test, have been performed to illustrate that our method works well for the interface tracing issue with high numerical accuracy. In addition, various representative qualitative tests have been presented to demonstrate the applicability of our method.

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.

Artificial compressibility approaches in flux reconstruction for incompressible viscous flow simulations

Artificial compressibility methods intend to offer divergence-free free velocity fields at the incompressible limit for compressible solvers. Three major approaches for this are compared within a high-order flux reconstruction framework: the established method (ACM) of Chorin (1967) and a new entropically damped method (EDAC) of Clausen (2013) which can keep velocity divergence sufficiently low to be run explicitly without the non-linear solver required by ACM. Furthermore, the ACM approach with hyperbolised diffusion is investigated. The accuracy and computational efficiency of these methods is investigated for a series of turbulent test cases over a range of Reynolds numbers. It is found for EDAC that velocity divergence scales linearly with the square root of compressibility, whereas for ACM a clear relation is not observed. EDAC is found to accurately resolve the low Reynolds number Taylor–Green vortex case; however, for the circular cylinder at Reynolds number 3900, earlier transition of the free shear-layer is observed due to an over-production of the turbulence kinetic energy. This over production of turbulent kinetic energy is attributed to the increased spatial pressure gradients of the EDAC method, and similar behaviour is observed for an aerofoil at Reynolds number 60 000 with an attached transitional boundary layer. These issues were not observed for the other ACM approaches. It is concluded that hyperbolic diffusion of ACM can be beneficial in terms of convergence but at the cost of case setup time, and EDAC can be a time efficient method for unsteady incompressible flows. However, care must be taken when reducing the stiffness of EDAC as the resulting pressure fluctuations can have a significant impact on transition.

Conditioned structure functions in turbulent hydrogen/air flames

Direct numerical simulation data obtained from two turbulent, lean hydrogen–air flames propagating in a box are analyzed to explore the influence of combustion-induced thermal expansion on turbulence in unburned gas. For this purpose, Helmholtz–Hodge decomposition is applied to the computed velocity fields. Subsequently, the second-order structure functions conditioned to unburned reactants are sampled from divergence-free solenoidal velocity field or irrotational potential velocity field, yielded by the decomposition. Results show that thermal expansion significantly affects the conditioned potential structure functions not only inside the mean flame brushes but also upstream of them. Upstream of the flames, first, transverse structure functions for transverse potential velocities grow with distance r between sampling points more slowly when compared to the counterpart structure functions sampled from the entire or solenoidal velocity field. Second, the former growth rate depends substantially on the distance from the flame-brush leading edge, even at small r. Third, potential root mean square (rms) velocities increase with the decrease in distance from the flame-brush leading edge and are comparable with solenoidal rms velocities near the leading edge. Fourth, although the conditioned axial and transverse potential rms velocities are always close to one another, thus implying isotropy of the potential velocity field in unburned reactants, the potential structure functions exhibit a high degree of anisotropy. Fifth, thermal expansion effects are substantial even for the solenoidal structure functions and even upstream of a highly turbulent flame. These findings call for development of advanced models of turbulence in flames, which allow for the discussed thermal expansion effects.

Non-uniqueness of weak solutions to 3D magnetohydrodynamic equations

We prove the non-uniqueness of weak solutions to 3D magnetohydrodynamic (MHD for short) equations. The constructed weak solutions do not conserve the magnetic helicity and can be close to any given smooth, divergence-free and mean-free velocity and magnetic fields. Furthermore, for any weak solutions in Ht,xβ˜ to the ideal MHD, where β˜>0, we prove that they are the strong vanishing viscosity and resistivity limit of the weak solutions to MHD equations. This shows that, in contrast to the weak ideal limits, Taylor's conjecture does not hold along the vanishing viscosity and resistivity limits. Inspired by the works on the NSE [1], ideal MHD [2] and transport equations [3], new types of velocity and magnetic flows, featuring both the refined spatial and temporal intermittency, are constructed to respect the geometry of MHD and to control the strong viscosity and resistivity. Compatible algebraic structure is derived in the convex integration scheme. More interestingly, the new intermittent flows indeed enable us to prove the aforementioned results for the hyper-viscous and hyper-resistive MHD equations up to the sharp exponent 5/4, which coincides exactly with the Lions exponent for 3D hyper-viscous NSE.

Extraction Of Temperature Fields From PIV Data Of Turbulent Rayleigh-Bénard Convection Using DNS

To gain insights into the dynamical processes that govern the problem of turbulent Rayleigh-Bénard convection, information about the temperature field---next to the velocity field---is required. In the present study, we exploit the governing flow equations to extract the temperature field from a velocity field measured via tomographic particle image velocimetry. The methods used within this study are derived from algorithms originally used in the framework of direct numerical simulations. First, we enforce the divergence-free condition on the measured velocity field using a fractional-step method based on a Poisson equation. Second, we extract the temperature field from the velocity field via the momentum equation. Third, we extract the temperature field from the velocity field via the solution of the stationary convection-diffusion equation. The extracted velocity field---together with the corresponding divergence-free velocity field---can serve as an initial condition for a direct numerical simulation of the problem. Validating the extraction method based on the stationary convection-diffusion equation with data from direct numerical simulations shows a good correlation coefficient between the extracted and the simulated temperature field (0.86). Although the third method incorporates the ambitious assumption of stationarity, temperature fields are successfully extracted from the measured velocity fields. However, with a correlation coefficient of 0.57, the agreement between the average extracted temperature field and the average temperature field measured in a subvolume of the flow domain is somewhat less than the one found in the simulated data. The latter is presumably attributed not only to the extraction method, but also to the underlying measurement data. Thus, an increase in measurement accuracy in future applications is likely to improve the accuracy of the extraction methods as well.

A phase-change model for diffusion-driven mass transfer problems in incompressible two-phase flows

We present a VOF-based numerical method for incompressible Direct Navier–Stokes (DNS) equations for diffusion-driven phase-change flows. A special emphasis is placed on the treatment of velocity discontinuities across the interface. A novel algorithm is presented to smoothly extend the liquid velocity field across the interface in a way that the interface can be transported by a divergence-free velocity field. The transport of species is treated with a two-scalar approach and special attention is paid to the advection and diffusion steps in order to prevent artificial mass transfer. The methodology is implemented in the open-source code Basilisk and is validated against analytical and semi-analytical models. The relative errors on the relevant quantities are generally below 1% for the finest grids. The method is finally applied to study the growth of electrochemically generated bubbles on planar electrodes and the effect of contact angles and number of nucleation sites is investigated.

A Semi-Lagrangian Computation of Front Speeds of G-equation in ABC and Kolmogorov Flows with Estimation via Ballistic Orbits

The Arnold-Beltrami-Childress (ABC) flow and the Kolmogorov flow are three dimensional periodic divergence free velocity fields that exhibit chaotic streamlines. We are interested in front speed enhancement in G-equation of turbulent combustion by large intensity ABC and Kolmogorov flows. We give a quantitative construction of the ballistic orbits of ABC and Kolmogorov flows, namely those with maximal large time asymptotic speeds in a coordinate direction. Thanks to the optimal control theory of G-equation (a convex but non-coercive Hamilton-Jacobi equation), the ballistic orbits serve as admissible trajectories for front speed estimates. To study the tightness of the estimates, we compute the front speeds of G-equation based on a semi-Lagrangian (SL) scheme with Strang splitting and weighted essentially non-oscillatory (WENO) interpolation. Time step size is chosen so that the Courant number grows sublinearly with the flow intensity. Numerical results show that the front speed growth rate in terms of the flow intensity may approach the analytical bounds from the ballistic orbits.

A new kinetic-energy-preserving method based on the convective rotational form

We present a new conservative discretization of the Navier-Stokes convective term that is also kinetic energy preserving (KEP). For divergence-free velocity fields, the specialized splitting recovers the traditional convective rotational form and therefore exhibits enhanced discrete secondary consistency with respect to vortical dynamics. The proposed method is first developed in the incompressible setting and is then extended to compressible flows via a square-root density weighting. It furthermore forms the basis for identifying a novel one-parameter family of KEP methods that leverages small-scale corrections to the momentum pressure gradient. Matrix-vector analysis associated with diagonal-norm finite difference summation-by-parts operators is used to straightforwardly study the preservation properties of different convective term formulations, including the new method, per the limitations of a discrete product rule. Preliminary evaluations on canonical test cases (i.e., isentropic and Taylor Green vortex simulations) are used to demonstrate the robustness of the current splitting and its performance with respect to representing secondary quantities.

Non-Gaussian limit of a tracer motion in an incompressible flow

We consider a massless tracer particle moving in a random, stationary, isotropic and divergence free velocity field. We identify a class of fields for which the limit of the laws of appropriately scaled tracer trajectory processes is a non-Gaussian Rosenb

Assimilation and extension of particle image velocimetry data of turbulent Rayleigh\u2013B\xe9nard convection using direct numerical simulations

A novel method for assimilating and extending measured turbulent Rayleigh–Bénard convection data is presented, which relies on the fractional step method also used to solve the incompressible Navier–Stokes equation in direct numerical simulations. Our approach is used to make measured tomographic particle image velocimetry (tomo PIV) fields divergence-free and to extract temperature fields. Comparing the time average of the extracted temperature fields with the temporally averaged temperature field, measured using particle image thermometry in a subdomain of the flow geometry, shows that extracted fields correlate well with measured fields with a correlation coefficient of C_{Ttilde{T}}=0.84. Additionally, extracted temperature fields as well as divergence-free velocity fields serve as initial fields for subsequent direct numerical simulations with and without feedback which generate small-scale turbulence initially absent in the experimental data. Although the tomo PIV data set was spatially under-resolved and did not include any information on the boundary layers, the here-proposed method successfully generates velocity and temperature fields featuring small-scale turbulence and thermal as well as kinetic boundary layers, without disturbing the large-scale circulation contained in the original experimental data significantly. The latter is underpinned by high vertical and horizontal velocity correlation coefficients—computed from velocity fields averaged in time and horizontal x-direction obtained from the measurement and from the simulation without feedback—of C_{vtilde{v}}=0.92 and C_{wtilde{w}}=0.91 representing the large-scale structure. For simulations with feedback, the generated velocity fields resemble the experimental data increasingly well for higher feedback gain values, whereas the temperature fluctuation intensity deviates noticeably from the values obtained from a direct numerical simulation without feedback for gain values alpha ge 1. Thus, a feedback gain of alpha =0.1 was found optimal with correlation coefficients of C_{vtilde{v}}=0.96 and C_{wtilde{w}}=0.95 as well as a realistic temperature fluctuation intensity profile. The xt-averaged temperature fields obtained from the direct numerical simulations with and without feedback correlate somewhat less with the extracted temperature field (C_{Ttilde{T}}approx 0.6), which is presumably caused by spatially under-resolved and temporally oscillating initial tomo PIV fields reflected by the extracted temperature field.Graphical abstract

Non-conservation of Dimension in Divergence-Free Solutions of Passive and Active Scalar Systems

For any \(h\in (1,2]\), we give an explicit construction of a compactly supported, uniformly continuous, and (weakly) divergence-free velocity field in \({\mathbb {R}}^2\) that weakly advects a measure whose support is initially the origin but for positive times has Hausdorff dimension h. These velocities are uniformly continuous in space-time and compactly supported, locally Lipschitz except at one point and satisfy the conditions for the existence and uniqueness of a Regular Lagrangian Flow in the sense of Di Perna and Lions theory. We then construct active scalar systems in \({\mathbb {R}}^2\) and \({\mathbb {R}}^3\) with measure-valued solutions whose initial support has co-dimension 2 but such that at positive times it only has co-dimension 1. The associated velocities are divergence free, compactly supported, continuous, and sufficiently regular to admit unique Regular Lagrangian Flows. This is in part motivated by the investigation of dimension conservation for the support of measure-valued solutions to active scalar systems. This question occurs in the study of vortex filaments in the three-dimensional Euler equations.

Triple Hill’s Vortex Synthetic Eddy Method

The generation of initial or inflow synthetic turbulent velocity or scalar fields reproducing statistical characteristics of realistic turbulence is still a challenge. The synthetic eddy method, previously introduced in the context of inflow conditions for large eddy simulations, is based on the assumption that turbulence can be regarded as a superposition of coherent structures. In this paper, a new type of synthetic eddy method is proposed, where the fundamental eddy is constructed by superposing three Hill's vortices, with their axes orthogonal to each other. A distribution of Hill's vortices is used to synthesize an anisotropic turbulent velocity field that satisfies the incompressibility condition and match a given Reynolds stress tensor. The amplitudes of the three vortices that form the fundamental eddy are calculated from known Reynolds stress profiles through a transformation from the physical reference frame to the principal-axis reference frame. In this way, divergence-free anisotropic turbulent velocity fields are obtained that can reproduce a given Reynolds stress tensor. The model was tested on both isotropic and anisotropic turbulent velocity fields, in the framework of grid turbulence decay and turbulent channel flow, respectively. The transition from artificial to realistic turbulence in the proximity to the inflow boundary was found to be small in all test cases that were considered.

Numerical investigation of the onset of three-dimensional characteristics in flow past a pair of square cylinders at various arrangements

An incompressible Navier–Stokes solver has been developed to analyze the three-dimensional viscous flow around a pair of square cylinders at a low Reynolds number (Re = 100) at various arrangements. A semi-implicit finite difference scheme was employed on a staggered grid, which facilitated the easier application of boundary conditions while achieving satisfactory accuracy and stability. A third-order upwind (Kawamura–Kuwahara) scheme was applied to discretize the convective terms and a second-order central difference scheme to discretize the diffusive terms. Time integration was performed using a second-order accurate Adams–Bashforth scheme. An iterative pressure–velocity correction was used to obtain a divergence-free velocity field. Aspects of the flow such as vorticity profiles, the time evolution of lift and drag coefficients, and three-dimensionality appearance were studied. Three-dimensional characteristics ceased to appear in a tandem arrangement, irrespective of the gap ratio. In contrast, the side by side arrangement showed a clear three-dimensional transition in the flow when analyzing the vortex structures. Further investigations carried out for staggered arrangement keeping the center to center distance constant and varying the stagger angle (α) reveal that the three-dimensional behavior starts at α = 30° and keeps progressing with the angle increase.