Flows In Unbounded Domains Research Articles

On the convergence rates of multi-dimensional subsonic irrotational flows in unbounded domains

On the convergence rates of multi-dimensional subsonic irrotational flows in unbounded domains

Efficient numerical method for Stokes flows in unbounded domains with informative boundary condition using axial Green function method

SummaryFlow calculations in an unbounded domain have limitations and challenges due to its infiniteness. A common approach is to impose a far‐field asymptotic condition to determine a unique flow. The leading behavior of the flow is identified at the far field, and then an unknown coefficient is assumed for the second behavior. This allows us to propose an efficient numerical method to solve two‐dimensional steady Stokes and potential flows in a truncated domain along with the coefficients. The second term provides crucial hydrodynamic information for the flow and is referred to as the informative boundary condition. The truncation creates artificial boundaries requiring boundary conditions for the approximate solution. The axial Green function method (AGM), combined with a specific one‐dimensional Green function over a semi‐infinite axis‐parallel line extended to infinity, allows us to implement the informative boundary condition in the truncated domain. AGMs, designed for complicated domains, are now applied to infinite domain cases because AGMs' versatility enables implementing the informative boundary condition by changing only the axial Green function. This approach's efficiency, accuracy, and consistency are investigated through several appealing Stokes flow problems including potential ones in infinite domains.

Unique solvability of elliptic problems associated with two-phase incompressible flows in unbounded domains

This paper shows the unique solvability of elliptic problems associated with two-phase incompressible flows, which are governed by the two-phase Navier-Stokes equations with a sharp moving interface, in unbounded domains such as the whole space separated by a compact interface and the whole space separated by a non-compact interface. As a by-product, we obtain the Helmholtz-Weyl decomposition for two-phase incompressible flows.

Towards filtered drag force model for non-cohesive and cohesive particle-gas flows

Euler-Lagrange simulations of gas-solid flows in unbounded domains have been performed to study sub-grid modeling of the filtered drag force for non-cohesive and cohesive particles. The filtered drag forces under various microstructures and flow conditions were analyzed in terms of various sub-grid quantities: the sub-grid drift velocity, which stems from the sub-grid correlation between the local fluid velocity and the local particle volume fraction, and the scalar variance of solid volume fraction, which is a measure to identify the degree of local inhomogeneity of volume fraction within a filter volume. The results show that the drift velocity and the scalar variance exert systematic effects on the filtered drag force. Effects of particle and domain sizes, gravitational accelerations, and mass loadings on the filtered drag are also studied, and it is shown that these effects can be captured by both sub-grid quantities. Additionally, the effect of cohesion force through the van der Waals interaction on the filtered drag force is investigated, and it is found that there is no significant difference on the dependence of the filtered drag coefficient of cohesive and non-cohesive particles on the sub-grid drift velocity or the scalar variance of solid volume fraction. The assessment of predictabilities of sub-grid quantities was performed by correlation coefficient analyses in a priori manner, and it is found that the drift velocity is superior. However, the drift velocity is not available in “coarse-grid” simulations and a specific closure is needed. A dynamic scale-similarity approach was used to model drift velocity but the predictability of that model is not entirely satisfactory. It is concluded that one must develop a more elaborate model for estimating the drift velocity in “coarse-grid” simulations.

Boundedness, almost periodicity and stability of certain Navier–Stokes flows in unbounded domains

We investigate the existence, uniqueness and stability of bounded and almost periodic mild solutions to several Navier–Stokes flow problems. In our strategy, we propose a general framework for studying the semi-linear evolution equations with certain smoothing properties of the linear part and with the local Lipschitz continuity of the nonlinear operator. Our method is based on interpolation functors combined with differential inequalities. Our abstract results are applied to Navier–Stokes–Oseen equations describing flows of incompressible viscous fluid passing a translating and rotating obstacle and to Navier–Stokes equations on aperture domains and/or in Besov spaces.

Computation of steady incompressible flows in unbounded domains

In this study we revisit the problem of computing steady Navier–Stokes flows in two-dimensional unbounded domains. Precise quantitative characterization of such flows in the high-Reynolds number limit remains an open problem of theoretical fluid dynamics. Following a review of key mathematical properties of such solutions related to the slow decay of the velocity field at large distances from the obstacle, we develop and carefully validate a spectrally-accurate computational approach which ensures the correct behavior of the solution at infinity. In the proposed method the numerical solution is defined on the entire unbounded domain without the need to truncate this domain to a finite box with some artificial boundary conditions prescribed at its boundaries. Since our approach relies on the streamfunction–vorticity formulation, the main complication is the presence of a discontinuity in the streamfunction field at infinity which is related to the slow decay of this field. We demonstrate how this difficulty can be overcome by reformulating the problem using a suitable background “skeleton” field expressed in terms of the corresponding Oseen flow combined with spectral filtering. The method is thoroughly validated for Reynolds numbers spanning two orders of magnitude with the results comparing favorably against known theoretical predictions and the data available in the literature.

Open boundary conditions for the Diffuse Interface Model in 1-D

New techniques are developed for solving multi-phase flows in unbounded domains using the Diffuse Interface Model in 1-D. They extend two open boundary conditions originally designed for the Navier–Stokes equations. The non-dimensional formulation of the DIM generalizes the approach to any fluid. The equations support a steady state whose analytical approximation close to the critical point depends only on temperature. This feature enables the use of detectors at the boundaries switching between conventional boundary conditions in bulk phases and a multi-phase strategy in interfacial regions. Moreover, the latter takes advantage of the steady state approximation to minimize the interface–boundary interactions. The techniques are applied to fluids experiencing a phase transition and where the interface between the phases travels through one of the boundaries. When the interface crossing the boundary is fully developed, the technique greatly improves results relative to cases where conventional boundary conditions can be used. Limitations appear when the interface crossing the boundary is not a stable equilibrium between the two phases: the terms responsible for creating the true balance between the phases perturb the interior solution. Both boundary conditions present good numerical stability properties: the error remains bounded when the initial conditions or the far field values are perturbed. For the PML, the influence of its main parameters on the global error is investigated to make a compromise between computational costs and maximum error. The approach can be extended to multiple spatial dimensions.

On calculation of hydrodynamic forces for steady flows in unbounded domains

This note addresses the question why the “impulse formula”, often employed to compute hydrodynamic forces in vortex-dominated time-dependent flows, is not applicable to steady flows in unbounded domains. By analyzing the asymptotic structure of steady and unsteady flow solutions in unbounded domains, it is demonstrated that one assumption made in the derivation of the impulse formula is in fact not satisfied in the steady case. This result also highlights the special character of steady flows in unbounded domains.

Global attractors of two-dimensional micropolar fluid flows in some unbounded domains

This paper is concerned with the existence and regularity of the global attractors of micropolar fluid flows in two-dimensional unbounded domains, in which the Poincaré inequality holds true. Based on an asymptotic compactness argument, a L 2 global attractor is shown to exist if the stationary external vector field is in H −1. Moreover, if the external vector field is in L 2, then the L 2 global attractor becomes an H 1 global attractor.

Asymptotic behavior of micropolar fluid flows

This paper is devoted to some recent results on several aspects of long time behavior of micropolar fluid flows. In particular, we consider such topics as existence and uniqueness of global in time solutions, their convergence to the stationary solution for large viscosity flows, existence of a global attractor and estimates of its Hausdorff and fractal dimensions, continuous dependence of solutions on microrotation viscosity perturbations and stability of the corresponding global attractor with respect to these perturbations, and flows in unbounded domains.

Steady-State Source Flow in Heterogeneous Porous Media

The flow of fluids in heterogeneous porous media is modelled by regarding the hydraulic conductivity as a stationary random space function. The flow variables, the pressure head and velocity field are random functions as well and we are interested primarily in calculating their mean values. The latter had been intensively studied in the past for flows uniform in the average. It has been shown that the average Darcy's law, which relates the mean pressure head gradient to the mean velocity, is given by a local linear relationship. As a result, the mean head and velocity satisfy the local flow equations in a fictitious homogeneous medium of effective conductivity. However, recent analysis has shown that for nonuniform flows the effective Darcy's law is determined by a nonlocal relationship of a convolution type. Hence, the average flow equations for the mean head are expressed as a linear integro-differential operator. Due to the linearity of the problem, it is useful to derive the mean head distribution for a flow by a source of unit discharge. This distribution represents a fundamental solution of the average flow equations and is called the mean Green function G d (x). The mean head G d(x) is derived here at first order in the logconductivity variance for an arbitrary correlation function ρ(x) and for any dimensionality d of the flow. It is obtained as a product of the solution G d (0)(x) for source flow in unbounded domain of the mean conductivity K A and the correction Ψ d (x) which depends on the medium heterogeneous structure. The correction Ψ d is evaluated for a few cases of interest. Simple one-quadrature expressions of Ψ d are derived for isotropic two- and three-dimensional media. The quadratures can be calculated analytically after specifying ρ (x) and closed form expressions are derived for exponential and Gaussian correlations. The flow toward a source in a three-dimensional heterogeneous medium of axisymmetric anisotropy is studied in detail by deriving Ψ3 as function of the distance from the source x and of the azimuthal angle θ. Its dependence on x, on the particular ρ(x) and on the anisotropy ratio is illustrated in the plane of isotropy (θ=0) and along the anisotropy axis (θ = π/2). The head factor k * is defined as a ratio of the head in the homogeneous medium to the mean head, k *=G d (0)/G d=Ψ d −1. It is shown that for isotropic conductivity and for any dimensionality of the flow the medium behaves as a one-dimensional and as an effective one close and far from the source, respectively, that is, lim x→0 k *(x) = K H/K A and lim x→∞ k *(x) = K efu/K A, where K A and K H are the arithmetic and harmonic conductivity means and K efu is the effective conductivity for uniform flow. For axisymmetric heterogeneity the far-distance limit depends on the direction. Thus, in the coordinate system of ρ(x) principal directions the limit values of k * are obtained as $$\begin{gathered} \tilde H_{{\text{cont}}}^ * (\mathcal{A},\mathcal{A}) = H(K_{\text{h}}^{{\text{efu}}} ) \subset _{{\text{diff}}}^ * (\mathcal{A},\mathcal{A}) = \mathcal{V}^ * ,{\text{ where }}\mathcal{A}\mathcal{U}\mathcal{R}q(\mathfrak{g})\mathfrak{k}\mathcal{U}_q (gl(2N + 1))\mathcal{R}\mathcal{B}(N) \subset \mathcal{U}_q \mathcal{F} \hfill \\ (K_{\text{h}}^{{\text{efu}}} K_{\text{v}}^{{\text{efu}}} )^{1/2} /K_{\text{A}} {\text{ for }}\theta = 0{\text{ and }}K_{\text{h}}^{{\text{efu}}} /K_{\text{A}} {\text{for }}\theta = \pi /2 \hfill \\ \end{gathered} $$ . These values differ from the corresponding components $$K_{\text{h}}^{{\text{efu}}} /K_{\text{A}} {\text{ and }}K_{\text{v}}^{{\text{efu}}} /K_{\text{A}} $$ of the effective conductivities tensor for uniform flow for θ = 0 and π/2, respectively. The results of the study are applied to solving the problem of the dipole well flow. The dependence of the mean head drop between the injection and production chambers on the anisotropy of the conductivity and the distance between the chambers is analyzed.

Unsteady 2-D Compressible Flow in Unbounded Domain

An unsteady isentropic flow model is presented to calculate the two-dimensional flow field in an arbitrarily closed region or in an open fluid domain. In the open domain, a unique boundary condition is implemented to simulate the infinite character of the open space. The characteristics-like method presented herein is shown to be robust over the entire subsonic flow range and, with the implementation of the infinite boundary, provides numerical results in agreement with analytic solutions and experimental data.

L2-decay for Navier-Stokes flows in unbounded domains, with application to exterior stationary flows

L2-decay for Navier-Stokes flows in unbounded domains, with application to exterior stationary flows

Navier-Stokes calculations for unsteady three-dimensional vortical flows in unbounded domains

Finite-difference Navier-Stokes calculations for unsteady, three-dimensional, incompressible, viscous flows induced by initial vorticity distributions are presented and discussed in this paper. The initial vorticity distributions are assumed to be embedded in a flow field of infinite extent that is quiescent at infinity. These vorticity distributions are typical of vortex rings and other closed vortical tubes or structures. Such structures are important elements in fluid flows such as jets, atmospheric convection and the far-field wakes of aircraft; studies of their interaction may aid in an understanding of complex fluid flows. The calculations employ a method recently proposed by Ting to approximate the infinite-domain boundary value problem with a finite boundary computational domain, and this method is shown to yield accurate three-dimensional results for reasonable expenditures of computer time. Because of the efficiency of the boundary condition technique and the resulting Navier-Stokes code, a 16-bit minicomputer with virtual memory was capable of performing the calculations for the unsteady motion of two obliquely colliding vortex rings. The results of these calculations are presented in the paper.

A sufficient condition for the instability of columnar vortices

The inviscid instability of columnar vortex flows in unbounded domains to three-dimensional perturbations is considered. The undisturbed flows may have axial and swirl velocity components with a general dependence on distance from the swirl axis. The equation governing the disturbance is found to simplify when the azimuthal wavenumber n is large. This permits us to develop the solution in an asymptotic expansion and reveals a class of unstable modes. The asymptotic results are confirmed by comparisons with numerical solutions of the full problem for a specific flow modelling the trailing vortex. It is found that the asymptotic theory predicts the most-unstable wave with reasonable accuracy for values of n as low as 3, and improves rapidly in accuracy as n increases. This study enables us to formulate a sufficient condition for the instability of columnar vortices as follows. Let the vortex have axial velocity W(r), azimuthal velocity V(r), where r is distance from the axis, let Ω be the angular velocity V/r, and let Γ be the circulation rV. Then the flow is unstable if $ V\frac{d\Omega}{dr}\left[ \frac{d\Omega}{dr}\frac{d\Gamma}{dr} + \left(\frac{dW}{dr}\right)^2\right] < 0.$

The weight function approach to uniqueness of viscous flows in unbounded domains

The weight function approach to uniqueness of viscous flows in unbounded domains

A uniqueness theorem for hydrodynamic flows in unbounded domains

We prove a uniqueness theorem for hydrodynamic motions in unbounded domains, which improves previous theorems.

On Uniqueness and Nonuniqueness in the Plane Creeping Flows of Second Order Fluids

The plane creeping flows of a second order incompressible viscoelastic fluid are considered and two theorems are derived to establish when the velocity field of such a fluid is identical to that of an incompressible Newtonian fluid. Application of the first theorem to three flows in unbounded domains is made and it is found that these three flows, respectively, have the same velocity fields as their Newtonian counterparts, and that these velocity fields are uniquely determined. The second theorem is used to show that when it is violated, “non-Newtonian velocity fields” may exist and an example is given, illustrating nonuniqueness in the velocity field.