Incompressible Fluid Dynamics Research Articles

A stopping criterion for the iterative solution of partial differential equations

A stopping criterion for iterative solution methods is presented that accurately estimates the solution error using low computational overhead. The proposed criterion uses information from prior solution changes to estimate the error. When the solution changes are noisy or stagnating it reverts to a less accurate but more robust, low-cost singular value estimate to approximate the error given the residual. This estimator can also be applied to iterative linear matrix solvers such as Krylov subspace or multigrid methods. Examples of the stopping criterion's ability to accurately estimate the non-linear and linear solution error are provided for a number of different test cases in incompressible fluid dynamics.

Simulations with current constraints of ELM-induced tungsten melt motion in ASDEX Upgrade

Melt motion simulations of recent ASDEX Upgrade experiments on transient-induced melting of a tungsten leading edge during ELMing H-mode are performed with the incompressible fluid dynamics code MEMOS 3D. The total current flowing through the sample was measured in these experiments providing an important constraint for the simulations since thermionic emission is considered to be responsible for the replacement current driving melt motion. To allow for a reliable comparison, the description of the space-charge limited regime of thermionic emission has been updated in the code. The effect of non-periodic aspects of the spatio-temporal heat flux in the temperature distribution and melt characteristics as well as the importance of current limitation are investigated. The results are compared with measurements of the total current and melt profile.

Effect of tilted electrostatic field on the Kelvin–Helmholtz instability in a liquid dielectric and gas flow

The effect of surface ponderomotive forces on the Kelvin–Helmholtz instability is studied in the linear formulation based on the equations and boundary conditions of the electrostatics and fluid dynamics of an ideal incompressible fluid. Conditions to be satisfied by the values of determining parameters of the problem for the transition of an unstable flow in zero electric field into a stable regime after the application of a horizontal electric field have been written in the form of inequalities. It has been shown that, at the stability bound, the wavelength of the most instable mode is independent of the ponderomotive forces. In case of a liquid with large permittivity a stable flow regime exists for which the stability condition only differs in small dimensionless values from the stability condition for the charged surface of a quiescent liquid conductor in contact with a gas at rest.

On some properties of the Navier–Stokes system of equations

We discuss the initial and boundary value problems for the system of dimensionless Navier–Stokes equations describing the dynamics of a viscous incompressible fluid using the method of characteristics and the geometric method developed by the authors. Some properties of the formulation of these problems are considered. We study the effect of the Reynolds number on the flow of a viscous fluid near the surface of a body.

Acoustic Scattering for Rotational and Translational Symmetric Structures in Nonuniform Potential Flow

An efficient approach is proposed to predict acoustic scattering with nonuniform potential flow effects for structures with rotational and translational symmetries. The convected wave equation is transformed to Helmholtz and Laplace equations using a time transformation. The boundary-element method is used to formulate scattering by rotationally symmetric structures as two separate block circulant matrix equations and, similarly, as two separate block Toeplitz matrix equations for structures with translational symmetry. Discrete Fourier transform is employed to solve the block circulant systems. The block Toeplitz systems are solved using the generalized minimal residual method along with the discrete Fourier transform. Solving the convected wave equation using structured matrices significantly reduces computational time and storage requirements. To demonstrate the application of the formulation, two exterior acoustic case studies are considered. The first case study examines acoustic scattering from a sphere submerged in potential flow under monopole source excitation. Directivity plots obtained using the proposed technique are compared with analytical results. The second case study examines flow-induced noise generated by a rigid cylinder immersed in low-Mach-number flow, with the effect of mean flow on the scattered acoustic field taken into account using nonuniform potential flow. The fluctuating flowfield is obtained using an incompressible computational fluid dynamics solver. Acoustic sources based on Lighthill’s analogy are extracted from the flowfield data using a high-order reconstruction scheme. Results from the hybrid computational fluid dynamics– boundary-element method technique are presented for turbulent flow past the cylinder, with Reynolds number based on cylinder diameter of and Mach number . The aeroacoustic results are compared with data from literature.

Computational reduction strategies for the detection of steady bifurcations in incompressible fluid-dynamics: Applications to Coanda effect in cardiology

We focus on reducing the computational costs associated with the hydrodynamic stability of solutions of the incompressible Navier–Stokes equations for a Newtonian and viscous fluid in contraction–expansion channels. In particular, we are interested in studying steady bifurcations, occurring when non-unique stable solutions appear as physical and/or geometric control parameters are varied. The formulation of the stability problem requires solving an eigenvalue problem for a partial differential operator. An alternative to this approach is the direct simulation of the flow to characterize the asymptotic behavior of the solution. Both approaches can be extremely expensive in terms of computational time. We propose to apply Reduced Order Modeling (ROM) techniques to reduce the demanding computational costs associated with the detection of a type of steady bifurcations in fluid dynamics. The application that motivated the present study is the onset of asymmetries (i.e., symmetry breaking bifurcation) in blood flow through a regurgitant mitral valve, depending on the Reynolds number and the regurgitant mitral valve orifice shape.

On the Application of Reduced Basis Methods to Bifurcation Problems in Incompressible Fluid Dynamics

In this paper we apply a reduced basis framework for the computation of flow bifurcation (and stability) problems in fluid dynamics. The proposed method aims at reducing the complexity and the computational time required for the construction of bifurcation and stability diagrams. The method is quite general since it can in principle be specialized to a wide class of nonlinear problems, but in this work we focus on an application in incompressible fluid dynamics at low Reynolds numbers. The validation of the reduced order model with the full order computation for a benchmark cavity flow problem is promising.

Computation of the self-noise of a controlled-diffusion airfoil based on the acoustic analogy

The self-noise of a controlled-diffusion airfoil is computed with several numerical techniques based on the acoustic analogy and involving different degrees of approximation. The flow solution is obtained through an incompressible large eddy simulation. The acoustic field as described by Lighthill’s analogy is computed with a finite element method applied to the exact airfoil geometry, and this solution is compared with results based on a half-plane Green’s function. This problem behaves as a classical trailing-edge noise problem for a wide range of frequencies; however, other mechanisms of sound production become significant at high frequencies. The results highlight the relative strengths and weaknesses of quadrupole- and dipole-based formulations of the acoustic analogy based on incompressible Computational Fluid Dynamics (CFD) results when applied to wall-bounded turbulent flows.

Approximate methods for equations of incompressible fluid

Approximate methods on the basis of sequential approximations in the theory of functional solutions to systems of conservation laws is considered, including the model of dynamics of incompressible fluid. Test calculations are performed, and a comparison with exact solutions is carried out.

A Modified Implementation of MINRES to Monitor Residual Subvector Norms for Block Systems

Saddle-point systems, i.e., structured linear systems with symmetric matrices are considered. A modified implementation of (preconditioned) MINRES is derived which allows subvectors of the residual to be monitored individually. Compared to the implementation from the textbook of [H. C. Elman, D. J. Silvester, and A. J. Wathen, Finite Elements and Fast Iterative Solvers: With Applications in Incompressible Fluid Dynamics, 2nd ed., Oxford University Press, Oxford, 2014], our method requires one extra vector of storage and no additional applications of the preconditioner. Numerical experiments are included.

Symmetries and conservation laws of the Euler equations in Lagrangian coordinates

We consider the Euler equations of incompressible inviscid fluid dynamics. We discuss a variational formulation of the governing equations in Lagrangian coordinates. We compute variational symmetries of the action functional and generate corresponding conservation laws in Lagrangian coordinates. We clarify and demonstrate relationships between symmetries and the classical balance laws of energy, linear momentum, center of mass, angular momentum, and the statement of vorticity advection. Using a newly obtained scaling symmetry, we obtain a new conservation law for the Euler equations in Lagrangian coordinates in n-dimensional space. The resulting integral balance relates the total kinetic energy to a new integral quantity defined in Lagrangian coordinates. This relationship implies an inequality which describes the radial deformation of the fluid, and shows the non-existence of time-periodic solutions with nonzero, finite energy.

Determination of Acoustic Impedance for Helmholtz Resonators Through Incompressible Unsteady Flow Simulations

The present work proposes and assesses a methodology based on incompressible computational fluid dynamics simulations to study the acoustic behavior of Helmholtz resonators under a large range of excitation amplitudes. It constitutes an alternative approach to the more widespread one based on compressible flow simulations to analyze the nonlinear regime of Helmholtz resonators. In the present methodology, the resonator is decomposed into its two main components: an assumed incompressible orifice neck and a compressible backing volume. The transfer impedance of the single orifice is obtained by means of an incompressible solver of the flow equations without turbulence modeling, whereas an analytical model accounts for the compliance of the gas in the backing cavity. The proposed methodology is compared for validation purposes to both numerical results of the full compressible equations and experimental data for the complete resonator at different sound pressure levels. A good agreement between the results of the two numerical approaches could be achieved. Numerical results match also fairly well with experimental data, but a systematic overprediction of the resistance by simulations is observed. Accounting for the presence of microrounded edges, presumably present due to manufacturing processes, allows a better agreement between numerical and experimental results.

ASSESSMENT OF THE EFFECT OF SPEED RATIOS IN NUMERICAL SIMULATIONS OF HIGHLY VISCOUS RUBBER MIXING IN A PARTIALLY FILLED CHAMBER

ABSTRACT Three-dimensional, transient, isothermal, and incompressible computational fluid dynamics (CFD) simulations are carried out for rubber mixing with two counter-rotating rotors in a partially filled chamber in order to assess the effect of different speed ratios. The three different speed ratios that are investigated include 1.0, 1.125, and 1.5. In addition to the solution of the incompressible continuity and momentum equations, a Eulerian multiphase model is employed to simulate two phases, rubber and air, and the volume of fluid (VOF) technique is used to calculate the free surface flow between the phases. The Bird–Carreau model is used to characterize the non-Newtonian highly viscous rubber. Massless particles are injected in the simulations to obtain data required for statistical calculations related to dispersive and distributive mixing characteristics. Specifically, joint probability density functions of mixing index and shear rate, and cumulative distribution functions of maximum shear stress are calculated to assess dispersive mixing, while distributive mixing capabilities are evaluated using various quantities such as cluster distribution index, axial distribution, interchamber particle transfer, and segregation scale. Results showed the speed ratio 1.125 to be consistently superior to 1.5 and 1.0, in terms of both dispersive and distributive mixing performance. The large speed difference between the rotors in the case of 1.5 caused it to perform the worst.

Coarse-graining two-dimensional turbulence via dynamical optimization

A model reduction technique based on an optimization principle is employed to coarse-grain inviscid, incompressible fluid dynamics in two dimensions. In this reduction the spectrally-truncated vorticity equation defines the microdynamics, while the macroscopic state space consists of quasi-equilibrium trial probability densities on the microscopic phase space, which are parameterized by the means and variances of the low modes of the vorticity. A macroscopic path therefore represents a coarse-grained approximation to the evolution of a nonequilibrium ensemble of microscopic solutions. Closure in terms of the vector of resolved variables, namely, the means and variances of the low modes, is achieved by minimizing over all feasible paths the time integral of their mean-squared residual with respect to the Liouville equation. The equations governing the optimal path are deduced from Hamilton–Jacobi theory. The coarse-grained dynamics derived by this optimization technique contains a scale-dependent eddy viscosity, modified nonlinear interactions between the low mode means, and a nonlinear coupling between the mean and variance of each low mode. The predictive skill of this optimal closure is validated quantitatively by comparing it against direct numerical simulations. These tests show that good agreement is achieved without adjusting any closure parameters.

Dynamics of the two-dimensional ideal incompressible fluid and Casimirs

Problems are considered in which the role of Casimirs in forming dynamics of the two-dimensional ideal incompressible fluid is basically studied; in particular, the conditions are formulated which arise in the stability problem of two-dimensional flows in the presence of Casimirs. Some general approaches to the construction of difference schemes for solving equations of two-dimensional fluid which possess the given Casimirs are considered.

Aeroacoustic Scattering Using a Particle Accelerated Computational Fluid Dynamics/Boundary Element Technique

A particle accelerated computational fluid dynamics/boundary element method technique to predict the sound pressure field produced by low Mach number flow past a rigid body is presented. An incompressible computational fluid dynamics solver is used to calculate the transient hydrodynamic flowfield. A near-field formulation based on Lighthill’s analogy is coupled with a particle condensation technique to predict the incident acoustic field and its normal derivative on the body. The near-field formulation involves singular surface and volume integrals, which are regularized via singularity subtraction. A particle condensation technique is applied to accelerate the incident field computations and reduce the amount of data that must be stored during the computational fluid dynamics analysis. The incident field is then combined with a boundary element method model of the body, and the scattered sound pressure field is obtained by solving the Burton–Miller boundary integral equations. The accuracy and computational cost of the particle accelerated computational fluid dynamics/boundary element method approach is demonstrated by calculating the incident acoustic field on a cylinder in crossflow under laminar and turbulent flow conditions as well as predicting the scattered and far-field sound pressures.

Concurrent finite element simulation of quadrupolar and dipolar flow noise in low Mach number aeroacoustics

The computation of flow-induced noise at low Mach numbers usually relies on a two-step hybrid methodolgy. In the first step, an incompressible fluid dynamics simulation (CFD) is performed and an acoustic source term is derived from it. The latter becomes the inhomogeneous term for an acoustic wave equation, which is solved in the second step, often resorting to boundary integral formulations. In the presence of rigid bodies, Curle’s acoustic analogy is probably the most extended approach. It has been shown that Curle’s boundary dipolar noise contribution does in fact correspond to the diffraction of the quadrupolar aerodynamic noise generated by the flow past the rigid body. In this work, advantage is taken from this fact to propose an alternative computational methodology to get the individual quadrupolar and dipolar contributions to the total acoustic pressure. For any linear acoustic wave operator, the unknown acoustic pressure can be split into its incident and diffracted components and be computed simultaneously to the incompressible flow field, in a single finite element computational run. This circumvents the problem found in Curle’s analogy of needing the total pressure at the body’s boundary, which includes the acoustic pressure fluctuations. The latter cannot be obtained from an incompressible CFD simulation. The proposed unified strategy could be beneficial for a large variety problems such as those involving noise generated from duct terminations, or those related with the simulation of fricatives in numerical voice production, among many others.

On the computation of measure-valued solutions

A standard paradigm for the existence of solutions in fluid dynamics is based on the construction of sequences of approximate solutions or approximate minimizers. This approach faces serious obstacles, most notably in multi-dimensional problems, where the persistence of oscillations at ever finer scales prevents compactness. Indeed, these oscillations are an indication, consistent with recent theoretical results, of the possible lack of existence/uniqueness of solutions within the standard framework of integrable functions. It is in this context that Young measures – parametrized probability measures which can describe the limits of such oscillatory sequences – offer the more general paradigm ofmeasure-valued solutionsfor these problems.We present viable numerical algorithms to compute approximate measure-valued solutions, based on the realization of approximate measures as laws of Monte Carlo sampled random fields. We prove convergence of these algorithms to measure-valued solutions for the equations of compressible and incompressible inviscid fluid dynamics, and present a large number of numerical experiments which provide convincing evidence for the viability of the new paradigm. We also discuss the use of these algorithms, and their extensions, in uncertainty quantification and contexts other than fluid dynamics, such as non-convex variational problems in materials science.

Non-splat singularity for the one-phase Muskat problem

For the water wave equations, the existence of splat singularities has been shown, i.e., the interface self-intersects along an arc in finite time. The aim of this paper is to show the absence of splat singularities for the incompressible fluid dynamics in porous media.

Physical characteristics of a lava flow determined from thermal measurements at the lava’s surface

We consider the problem about determination of characteristics of a lava flow from the physical parameters measured on its surface. The problem is formulated as an inverse boundary problem for the model simulating the dynamics of a viscous heat-conducting incompressible inhomogeneous fluid, where, on the basis of additional data at one part of the model boundary, the missing conditions at another part of the boundary have to be determined, and then the characteristics of fluid in the entire model domain have to be reconstructed. The considered problem is ill-posed. We develop a numerical approach to the solution of the problem in the case of a steady-state flow. Assuming that the temperature and the heat flow are known at the upper surface of the lava, we determine the flow characteristics inside the lava. We compute model examples and show that the lava temperature and flow velocity can be determined with a high precision when the initial data are smooth or slightly noisy.