Solenoidal Velocity Field Research Articles

Backscatter of scalar variance in turbulent premixed flames

To explore the direction of inter-scale transfer of scalar variance between subgrid scale (SGS) and resolved scalar fields, direct numerical simulation data obtained earlier from two complex-chemistry lean hydrogen–air flames are analysed by applying Helmholtz–Hodge decomposition (HHD) to the simulated velocity fields. Computed results show backscatter of scalar (combustion progress variable$c$) variance, i.e. its transfer from SGS to resolved scales, even in a highly turbulent flame characterized by a unity-order Damköhler number and a ratio of Kolmogorov length scale to thermal laminar flame thickness as low as 0.05. Analysis of scalar fluxes associated with the solenoidal and potential velocity fields yielded by HHD shows that the documented backscatter stems primarily from the potential velocity perturbations generated due to dilatation in instantaneous local flames, with the backscatter being substantially promoted by a close alignment of the spatial gradient of mean scalar progress variable and the potential-velocity contribution to the local SGS scalar flux. The alignment is associated with the fact that combustion-induced thermal expansion increases local velocity in the direction of$\boldsymbol {\nabla } c$. These results call for development of SGS models capable of predicting backscatter of scalar variance in turbulent flames in large eddy simulations.

Meshfree Semi-Lagrangian Methods for Solving Surface Advection PDEs

We analyze a class of meshfree semi-Lagrangian methods for solving advection problems on smooth, closed surfaces with solenoidal velocity field. In particular, we prove the existence of an embedding equation whose corresponding semi-Lagrangian methods yield the ones in the literature for solving problems on surfaces. Our analysis allows us to apply standard bulk domain convergence theories to the surface counterparts. In addition, we provide detailed descriptions for implementing the proposed methods to run on point clouds. After verifying the convergence rates against the theory, we show that the proposed method is a robust building block for more complicated problems, such as advection problems with non-solenoidal velocity field, inviscid Burgers’ equations and systems of reaction advection diffusion equations for pattern formation.

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.

Quasi-stationarity of scalar turbulent mixing statistics in a non-symmetric case

The existence of an asymptotic shape at large times of the probability density function (PDF) of a non-reacting scalar, mixed by a solenoidal turbulent velocity field and molecular diffusive transport, was investigated by Sinai and Yakhot [Y. G. Sinai and V. Yakhot, “Limiting probability distributions of a passive scalar in a random velocity field,” Phys. Rev. Lett. 63, 1962 (1989)]. The quasi-stationarity of the mixing statistics along the time evolution by Valiño et al. [“Quasistationary probability density functions in the turbulent mixing of a scalar field,” Phys. Rev. Lett. 72, 3518 (1994)] was an extension to symmetric scalar pdfs; analytic solutions for the scalar fluctuation dissipation rates, conditional upon the scalar value, and the pdf were obtained. This manuscript examines the generalization of the latter results to asymmetric scalar pdfs, further scrutinizes underlying mechanisms for a quasi-stationary statistics, and shows a Monte Carlo implementation which allows non-Gaussian relaxations to prescribed values of skweness and kurtosis.

Unravelling cosmic velocity flows: a Helmholtz–Hodge decomposition algorithm for cosmological simulations

In the context of intra-cluster medium turbulence, it is essential to be able to split the turbulent velocity field in a compressive and a solenoidal component. We describe and implement a new method for this aim, i.e., performing a Helmholtz–Hodge decomposition, in multi-grid, multi-resolution descriptions, focusing on (but not being restricted to) the outputs of AMR cosmological simulations. The method is based on solving elliptic equations for a scalar and a vector potential, from which the compressive and the solenoidal velocity fields, respectively, are derived through differentiation. These equations are addressed using a combination of Fourier (for the base grid) and iterative (for the refinement grids) methods. We present several idealised tests for our implementation, reporting typical median errors in the order of 1‰-1%, and with 95-percentile errors below a few percents. Additionally, we also apply the code to the outcomes of a cosmological simulation, achieving similar accuracy at all resolutions, even in the case of highly non-linear velocity fields. We finally take a closer look to the decomposition of the velocity field around a massive galaxy cluster.

Local well-posedness of a quasi-incompressible two-phase flow

We show well-posedness of a diffuse interface model for a two-phase flow of two viscous incompressible fluids with different densities locally in time. The model leads to an inhomogeneous Navier–Stokes/Cahn–Hilliard system with a solenoidal velocity field for the mixture, but a variable density of the fluid mixture in the Navier–Stokes type equation. We prove existence of strong solutions locally in time with the aid of a suitable linearization and a contraction mapping argument. To this end, we show maximal L^2-regularity for the Stokes part of the linearized system and use maximal L^p-regularity for the linearized Cahn–Hilliard system.

Solenoidal and potential velocity fields in weakly turbulent premixed flames

Direct Numerical Simulation data obtained earlier from two statistically 1D, planar, fully-developed, weakly turbulent, single-step-chemistry, premixed flames characterized by different (7.53 and 2.50) density ratios σ are analyzed to explore the influence of combustion-induced thermal expansion on the turbulence and the backward influence of such flow perturbations on the reaction zone. For this purpose, the actual velocity fields are decomposed into solenoidal and potential velocity subfields, followed by independently processing each subfield. Results show that the potential and solenoidal velocities are negatively (positively) correlated in unburned reactants (burned products, respectively) provided that σ=7.53. Moreover, not only potential, but also solenoidal velocity subfields contribute to countergradient turbulent scalar transport. Furthermore, correlation between strain rates generated by the solenoidal and potential velocity fields and conditioned to the reaction zone is positive (negative) in the leading (trailing, respectively) halves of the mean flame brushes. The potential strain rate correlates negatively with the flame curvature within the reaction zone, whereas the solenoidal strain rate and the curvature are negatively (positively) correlated in the leading (trailing, respectively) halves of the mean flame brushes. If σ=7.53, the stretch rate conditioned to the reaction zone is negative (positive) in regions characterized by large positive (negative, respectively) potential strain rates, whereas the opposite trend is observed for the solenoidal strain rate. Thus, the potential and solenoidal velocity fields differently affect the reaction zone. In the case of σ=2.5, such differences are significantly less pronounced. Finally, an approximate decomposition of the mean rate of viscous dissipation of flow kinetic energy into solenoidal and potential contributions is suggested and supported by DNS data. The results also show that the dilatation-induced dissipation plays an important role. Therefore, the influence of thermal expansion on the dissipation rate is not reduced to an increase in mixture viscosity in the rotational dissipation rate.

Application of Helmholtz-Hodge decomposition and conditioned structure functions to exploring influence of premixed combustion on turbulence upstream of the flame

In order to explore the influence of combustion-induced thermal expansion on turbulence, a new research method is introduced. The method consists in jointly applying Helmholtz-Hodge decomposition and conditioned structure functions to analyzing turbulent velocity fields. Opportunities offered by the method are demonstrated by using it to process Direct Numerical Simulation data obtained earlier from two statistically 1D, planar, fully-developed, weakly turbulent, single-step-chemistry, premixed flames characterized by two significantly different (7.52 and 2.50) density ratios, with all other things being approximately equal. To emphasize the influence of combustion-induced thermal expansion on turbulent flow of unburned mixture upstream of a premixed flame, the focus of analysis is placed on structure functions conditioned to the unburned mixture in both points. Two decomposition techniques, i.e. (i) a widely used orthogonal Helmholtz-Hodge decomposition and (ii) a recently introduced natural Helmholtz-Hodge decomposition, are probed, with results obtained using them being similar in the largest part of the computational domain with the exception of narrow zones near the inlet and outlet boundaries. Computed results indicate that combustion-induced thermal expansion can significantly change turbulent flow of unburned mixture upstream of a premixed flame by generating anisotropic potential velocity fluctuations whose spatial structure differ substantially from spatial structure of the incoming turbulence. The magnitude of such potential velocity fluctuations is greater than the magnitude of the solenoidal velocity fluctuations in the largest part of the mean flame brush in the case of the high density ratio. In the case of the low density ratio, the latter magnitude is larger everywhere, but the two magnitudes are comparable in the middle of the mean flame brush. Contrary to the potential velocity fluctuations, the influence of the thermal expansion on the solenoidal velocity field in the unburned mixture is of minor importance under conditions of the present study.

Gabor mode enrichment in large eddy simulations of turbulent flow

Abstract

Macroscopic models for filtration and heterogeneous reactions in porous media

Derivation of macroscopic models for advection-diffusion processes in the presence of dominant heterogeneous (e.g., surface) reactions using homogenisation theory or volume averaging is often deemed unfeasible (Valdés-Parada et al., 2011; Battiato and Tartakovsky, 2011a) due to the strong coupling between scales that characterise such systems. In this work, we show how the upscaling can be carried out by applying and extending the methods presented in Allaire and Raphael (2007), Mauri (1991). The approach relies on the decomposition of the microscale concentration into a reactive component, given by the eigenfunction of the advection-diffusion operator, the associated eigenvalue which represents the macroscopic effective reaction rate, and a non-reactive component. The latter can be then upscaled with a two-scale asymptotic expansion and the final macroscopic equation is obtained for the leading order. The same method can also be used to overcome another classical assumption, namely of non solenoidal velocity fields, such as the case of deposition of charged colloidal particles driven by electrostatic potential forces. The whole upscaling procedure, which consists in solving three cell problems, is implemented for arbitrarily complex two- and three-dimensional periodic structures using the open-source finite volume library OpenFOAM®. We provide details on the implementation and test the methodology for two-dimensional periodic arrays of spheres, and we compare the results against fully resolved numerical simulations, demonstrating the accuracy and generality of the upscaling approach. The effective velocity, dispersion and reaction coefficients are obtained for a wide range of Péclet and surface Damköhler numbers, and for Coulomb-like forces to the grains. Noticeably, all the effective transport parameters are significantly different from the non-reactive (conserved scalar) case, as the heterogeneity introduced by the reaction strongly affects the micro-scale profiles.

Generation of Solenoidal Modes and Magnetic Fields in Turbulence Driven by Compressive Driving

Abstract We perform numerical simulations of hydrodynamic (HD) and magnetohydrodynamic (MHD) turbulence driven by compressive driving, to study the generation of solenoidal velocity components and the small-scale magnetic field. We mainly focus on the effects of mean magnetic field (B 0) and the sonic Mach number (M s ). The dependence of solenoidal ratio (i.e., ratio of solenoidal to kinetic energies) and magnetic energy density on M s in compressively driven turbulence is already established, but that on B 0 is not yet. We also consider two different driving schemes in terms of the correlation timescale of forcing vectors: a finite-correlated driving and a delta-correlated driving. Our findings are as follows. First, when we fix the value of B 0, the solenoidal ratio after saturation increases as increases. A similar trend is observed for generation of magnetic field when B 0 is small. Second, when we fix the value of , HD and MHD simulations result in similar solenoidal ratios when B 0 is not strong (say, M A ≳ 5, where M A is Alfvén Mach number). However, the ratio increases when M A ≲ 5. Roughly speaking, the magnetic energy density after saturation is a linearly increasing function of B 0 irrespective of M s . Third, generation of the solenoidal velocity component is not sensitive to numerical resolution, but that of magnetic energy density is mildly sensitive. Finally, when initial conditions are same, the finite-correlated driving always produces more solenoidal velocity and small-scale magnetic field components than the delta-correlated driving. We additionally analyze the vorticity equation to understand why higher and B 0 yield a larger quantity of the solenoidal velocity component.

Existence and asymptotic results for an intrinsic model of small-strain incompatible elasticity

A general model of incompatible small-strain elasticity is presented and analyzed, based on the linearized strain and its associated incompatibility tensor field. Strain incompatibility accounts for the presence of dislocations, whose motion is ultimately responsible for the plastic behaviour of solids. The specific functional setting is built up, on which existence results are proved. Our solution strategy is essentially based on the projection of the governing equations on appropriate subspaces in the spirit of the Leray decomposition of solenoidal square-integrable velocity fields in hydrodynamics. It is also strongly related with the Beltrami decomposition of symmetric tensor fields in the wake of previous works by the authors. Moreover a novel model parameter is introduced, the incompatibility modulus, that measures the resistance of the elastic material to incompatible deformations. An important result of our study is that classical linearized elasticity is recovered as the limit case when the incompatibility modulus goes to infinity. Several examples are provided to illustrate this property and the physical meaning of the incompatibility modulus in connection with the dissipative nature of the processes under consideration.

Two-dimensional isotropic turbulent inflow conditions for vortex particle method

The unsteady loading on an airfoil due to the inflow turbulence is the primary source of broadband noise in turbomachinery, which makes the modeling of the inflow conditions a crucial task. This Rapid Communication presents an advanced method to model isotropic turbulent inflow conditions for a two-dimensional vortex particle method. The method relies on the classical view of turbulence as a superposition of random vortices convected with the mean flow. The fluctuating velocity is formulated using the scalar potential given by a convolution product which furnishes a solenoidal velocity field. The system is then optimized stochastically to create a realistic turbulence field based on a target energy spectra using an evolutionary optimization algorithm.Received 27 July 2018DOI:https://doi.org/10.1103/PhysRevFluids.4.022701©2019 American Physical SocietyPhysics Subject Headings (PhySH)Research AreasAerodynamic noiseVortex flowsTechniquesTurbulenceFluid Dynamics

A Hybrid Analytics Paradigm Combining Physics-Based Modeling and Data-Driven Modeling to Accelerate Incompressible Flow Solvers

Numerical solution of the incompressible Navier–Stokes equations poses a significant computational challenge due to the solenoidal velocity field constraint. In most computational modeling frameworks, this divergence-free constraint requires the solution of a Poisson equation at every step of the underlying time integration algorithm, which constitutes the major component of the computational expense. In this study, we propose a hybrid analytics procedure combining a data-driven approach with a physics-based simulation technique to accelerate the computation of incompressible flows. In our approach, proper orthogonal basis functions are generated to be used in solving the Poisson equation in a reduced order space. Since the time integration of the advection–diffusion equation part of the physics-based model is computationally inexpensive in a typical incompressible flow solver, it is retained in the full order space to represent the dynamics more accurately. Encoder and decoder interface conditions are provided by incorporating the elliptic constraint along with the data exchange between the full order and reduced order spaces. We investigate the feasibility of the proposed method by solving the Taylor–Green vortex decaying problem, and it is found that a remarkable speed-up can be achieved while retaining a similar accuracy with respect to the full order model.

A divergence-free low-order stabilized finite element method for a generalized steady state Boussinesq problem

In this work we propose and analyze a new stabilized finite element method for the coupled Navier–Stokes/temperature (or Boussinesq) equations. The method is built using low order conforming elements for velocity and temperature, and piecewise constant elements for pressure. With the help of the lowest order Raviart–Thomas space, a lifting of the jumps of the discrete pressure is built in such a way that when this lifting is added to the conforming velocity field, the resulting velocity is solenoidal (at the price of being non-conforming). This field is then fed to the momentum and temperature equations, guaranteeing that the convective terms in these equations are antisymmetric, without the need of altering them, thus simplifying the analysis of the resulting method. Existence of solutions, discrete stability, and optimal convergence are proved for both the conforming velocity field, and its corresponding divergence-free non-conforming counterpart. Numerical results confirm the theoretical findings, as well as the gain provided by the solenoidal discrete velocity field over the conforming one.

Numerical simulation of the planar extrudate swell of pseudoplastic and viscoelastic fluids with the streamfunction and the VOF methods

We present an Eulerian free-surface flow solver for incompressible pseudoplastic and viscoelastic non-Newtonian fluids. The free-surface flow solver is based on the streamfunction flow formulation and the volume-of-fluid method. The streamfunction solver computes the vector potential of a solenoidal velocity field, which ensures by construction the mass conservation of the solution, and removes the pressure unknown. Pseudoplastic liquids are modelled with a Carreau model. The viscoelastic fluids are governed by differential constitutive models reformulated with the log-conformation approach, in order to preserve the positive-definiteness of the conformation tensor, and to circumvent the high Weissenberg number problem. The volume fraction of the fluid is advected with a geometric conservative unsplit scheme that preserves a sharp interface representation. For the sake of comparison, we also implemented an algebraic advection scheme for the liquid volume fraction. The proposed numerical method is tested by simulating the planar extrudate swell with the Carreau, Oldroyd-B and Giesekus constitutive models. The swell ratio of the extrudates are compared with the data available in the literature, as well as with numerical simulations performed with the open-source rheoTool toolbox in OpenFOAM®. While the simulations of the generalized Newtonian fluids achieved mesh independence for all the methods tested, the flow simulations of the viscoelastic fluids are more sensitive to mesh refinement and the choice of numerical scheme. Moreover, the simulations of Oldroyd-B fluid flows above a critical Weissenberg number are prone to artificial surface instabilities. These numerical artifacts are due to discretization errors within the Eulerian surface-capturing method. However, the numerical issues arise from the stress singularity at the die exit corner, and the unphysical predictions of the Oldroyd-B model in the skin layer of the extrudate after the die exit, where large extensional deformations occur.

A solution class of the Euler equation in a torus with solenoidal velocity field. II

A solution class of the Euler equation in a torus with solenoidal velocity field. II

A vorticity based approach to handle the fluid-structure interaction problems

A vorticity based approach for the numerical solution of the fluid-structure interaction problems is introduced in which the fluid and structure(s) can be viewed as a continuum. Retrieving the vorticity field and recalculating a solenoidal velocity field, specially at the fluid-structure interface, are the kernel of the proposed algorithm. In the suggested method, a variety of constitutive equations as a function of left Cauchy–Green deformation tensor can be applied for modeling the structure domain. A nonlinear Mooney–Rivlin and Saint Venant–Kirchhoff model are expressed in terms of the left Cauchy–Green deformation tensor and the presented method is able to model the behavior of a visco-hyperelastic structure in the incompressible flow. Some numerical experiments, with considering the neo-Hookean model for structure domain, are executed and the results are validated via the available results from literature.

A solution class of the Euler equation in a torus with solenoidal velocity field. III

A solution class of the Euler equation in a torus with solenoidal velocity field. III

Solenoidal filtering of volumetric velocity measurements using Gaussian process regression

Volumetric velocity measurements of incompressible flows contain spurious divergence due to measurement noise, despite mass conservation dictating that the velocity field must be divergence-free (solenoidal). We investigate the use of Gaussian process regression to filter spurious divergence, returning analytically solenoidal velocity fields. We denote the filter solenoidal Gaussian process regression (SGPR) and formulate it within the Bayesian framework to allow a natural inclusion of measurement uncertainty. To enable efficient handling of large data sets on regular and near-regular grids, we propose a solution procedure that exploits the Toeplitz structure of the system matrix. We apply SGPR to two synthetic and two experimental test cases and compare it with two other recently proposed solenoidal filters. For the synthetic test cases, we find that SGPR consistently returns more accurate velocity, vorticity and pressure fields. From the experimental test cases, we draw two important conclusions. Firstly, it is found that including an accurate model for the local measurement uncertainty further improves the accuracy of the velocity field reconstructed with SGPR. Secondly, it is found that all solenoidal filters result in an improved reconstruction of the pressure field, as verified with microphone measurements. The results obtained with SGPR are insensitive to correlation length, demonstrating the robustness of the filter to its parameters.