Classical Navier Research Articles

Zonal Detached-Eddy Simulation of Separated Flow Around a Finite-Span Wing

The unsteady turbulent flow around a three-dimensional finite-span wing at stall is investigated by means of a zonal detached-eddy simulation. The computation captures a nondelayed development of unsteady structures at the beginning of the separation and accurately resolves the energy cascade in the fully turbulent mixing layer over a large frequency range. A comparison with the experimental data shows that the zonal detached-eddy simulation approach clearly improves the prediction of the mean flow, whereas classical Reynolds-averaged Navier–Stokes methods usually fail. A spectral analysis of pressure signals highlights the origin and the characteristics of the various unsteady mechanisms involved in this complex flow.

Mathematical models and numerical methods for the simulation of adaptive inflatable structures for impact absorption

The paper describes various approaches for the mathematical modelling of Adaptive Inflatable Structures (AIS) along with the corresponding numerical methods. The introductory part presents a general idea of adaptive impact absorption (AIA) and the concept of inflatable structures equipped with controllable valves serving for internal pressure control. Application of AIS for adaptive absorption of the impact loading is briefly explained. The paper focuses on diverse methods of modelling of inflatable structures, which are based on interaction between solid walls and fluid enclosed inside. Modelling of the solid walls is based on rigid body dynamics or initial-boundary value problem of solid mechanics. In turn, modelling of the fluid utilizes either classical equilibrium thermodynamics or Navier–Stokes equations. Consequently, four possible combinations of the above approaches are distinguished, precisely analyzed and applied for the modelling of different types of inflatable structures. Each model takes into account controllable valves, which requires introducing additional coupling between parameters defining the valves and selected results of the analysis. Corresponding numerical methods include classical methods of solving ordinary differential equations, finite volume method (FVM) applied for problems with mobile boundaries, finite element method (FEM) applied for problems involving additional ODEs and, finally, FEM coupled with FVM. Proposed numerical methods and software tools are utilized for the simulation of adaptive pneumatic cylinders, adaptive pneumatic fenders and membrane valves.

Global smooth axisymmetric solutions of 3-D inhomogeneous incompressible Navier–Stokes system

In this paper, we investigate the global regularity to 3-D inhomogeneous incompressible Navier–Stokes system with axisymmetric initial data which does not have swirl component for the initial velocity. We first prove that the \(L^\infty \) norm to the quotient of the inhomogeneity by r, namely \(a/r\buildrel \hbox {def}\over =(1/\rho -1)/r,\) controls the regularity of the solutions. Then we prove the global regularity of such solutions provided that the \(L^\infty \) norm of \(a_0/r\) is sufficiently small. Finally, with additional assumption that the initial velocity belongs to \(L^p\) for some \(p\in [1,2),\) we prove that the velocity field decays to zero with exactly the same rate as the classical Navier–Stokes system.

Traveling wave solutions to incompressible unsteady 2-D laminar flows with heat transfer boundary

Analytical solutions play important roles in the understanding of fluid dynamics and heat transfer related problems. Some analytical solutions for incompressible steady/unsteady 2-D problems have been obtained in literature, but only a few of those are found under heat transfer conditions (which brings more complexities into the problem). This paper is focused on the analytical solutions to the basic problem of incompressible unsteady 2-D laminar flows with heat transfer. By using the traveling wave method, fluid dynamic governing equations are developed based on classical Navier–Stokes equations and can be reduced to ordinary differential equations, which provide reliable explanations to the 2-D fluid flows. In this study, a set of analytical solutions to incompressible unsteady 2-D laminar flows with heat transfer are obtained. The results show that both the velocity field and the temperature field take an exponential function form, or a polynomial function form, when traveling wave kind solution is assumed and compared in such fluid flow systems. In addition to heat transfer problem, the effects of boundary input parameters and their categorization and generalization of field forming or field evolutions are also obtained in this study. The current results are also compared with the results of Cai et al. (R. X. Cai, N. Zhang. International Journal of Heat and Mass Transfer, 2002, 45: 2623-2627) and others using different methods. It is found that the current method can cover the results and will also extend the fluid dynamic model into a much wider parameter ranges (and flow situations).

From the Euler–Bernoulli beam to the Timoshenko one through a sequence of Reddy-type shear deformable beam models of increasing order

Abstract A sequence of elastic Reddy-type shear deformable beams of increasing (odd) order is envisioned, which starts with the Euler–Bernoulli beam (first order) and terminates with the Timoshenko beam (infinite order). The kinematics of the generic beam, including the warping mode of the cross sections, is specified in terms of three deformation variables (two curvatures, one shear angle), work-conjugate of as many stress resultants (two bending moments, one shear force). The principle of virtual power is used to determine the (static) equilibrium equations and the boundary conditions. The equations relating the bending moments and shear force to the curvatures and shear angle are also reported. Two governing differential equations (equivalent to a deflection differential equation of the sixth order) are provided, along with their general solution to within six unknown constants; the solutions for the Euler–Bernoulli and Timoshenko beams are recovered for first and infinite orders, respectively. A principle of minimum total potential energy is shown to hold for the generic beam. Extended forms of the classical Navier formula for the computation of the (axial) normal stress, and of the Jourawski formula for the computation of the “equilibrium” shear stress, are provided. A simple cantilever beam is used for an illustrative application.

Global Solutions of a Viscous Gas-Liquid Model with Unequal Fluid Velocities in a Closed Conduit

We consider a compressible gas-liquid drift-flux model with a general slip law commonly used to describe realistic two-phase flow scenarios. The slip law will introduce a difference in the magnitude of the two fluid velocities, and they possibly will also have different sign. This allows the model to describe the effect of buoyant forces, for example, in a vertical conduit, where heavy liquid will move downward due to gravity whereas light gas will be displaced upwardly. Combining the two mass equations and the mixture momentum equation with the slip law, the model can be expressed in terms of the gas-fluid velocity and a generalized pressure function that depends on the common pressure and three new terms that depend on the liquid mass and the gas velocity. This generalized pressure function introduces new nonlinear effects and coupling mechanisms between the two mass equations and the mixture momentum equation which require careful refinements of techniques previously used for the analysis of the classical Navier--Stokes model. In this work we discuss global existence and uniqueness of strong solutions considered in a closed one-dimensional conduit subject to appropriate smallness assumptions and regularity assumptions on the initial data. The analysis provides insight into the special role played by the slip parameters. This work presents a first global existence result for the drift-flux model with a general slip law.

Finite Element Modeling of Free Surface Flow in Variable Porosity Media

The aim of the present work is to present an overview of some numerical procedures for the simulation of free surface flows within a porous structure. A particular algorithm developed by the authors for solving this type of problems is presented. A modified form of the classical Navier–Stokes equations is proposed, with the principal aim of simulating in a unified way the seepage flow inside rockfill-like porous material and the free surface flow in the clear fluid region. The problem is solved using a semi-explicit stabilized fractional step algorithm where velocity is calculated using a 4th order Runge–Kutta scheme. The numerical formulation is developed in an Eulerian framework using a level set technique to track the evolution of the free surface. An edge-based data structure is employed to allow an easy OpenMP parallelization of the resulting finite element code. The numerical model is validated against laboratory experiments on small scale rockfill dams and is compared with other existing methods for solving similar problems.

A data-assimilation method for Reynolds-averaged Navier–Stokes-driven mean flow reconstruction

AbstractWe present a data-assimilation technique based on a variational formulation and a Lagrange multipliers approach to enforce the Navier–Stokes equations. A general operator (referred to as the measure operator) is defined in order to mathematically describe an experimental measure. The presented method is applied to the case of mean flow measurements. Such a flow can be described by the Reynolds-averaged Navier–Stokes (RANS) equations, which can be formulated as the classical Navier–Stokes equations driven by a forcing term involving the Reynolds stresses. The stress term is an unknown of the equations and is thus chosen as the control parameter in our study. The data-assimilation algorithm is derived to minimize the error between a mean flow measurement and the measure performed on a numerical solution of the steady, forced Navier–Stokes equations; the optimal forcing is found when this error is minimal. We demonstrate the developed data-assimilation framework on a test case: the two-dimensional flow around an infinite cylinder at a Reynolds number of $\mathit{Re}=150$. The mean flow is computed by time-averaging instantaneous flow fields from a direct numerical simulation (DNS). We then perform several ‘measures’ on this mean flow and apply the data-assimilation method to reconstruct the full mean flow field. Spatial interpolation, extrapolation, state vector reconstruction and noise filtering are considered independently. The efficacy of the developed identification algorithm is quantified for each of these cases and compared with more traditional methods when possible. We also analyse the identified forcing in terms of unsteadiness characterization, present a way to recover the second-order statistical moments of the fluctuating velocities and finally explore the possibility of pressure reconstruction from velocity measurements.

Numerical simulation of particle formation in the rapid expansion of supercritical solution process

In this paper, we numerically study particle formation in the rapid expansion of supercritical solution (RESS) process in a two dimensional, axisymmetric geometry, for a benzoic acid+CO2 system. The fluid is described by the classical Navier–Stokes equation, with the thermodynamic pressure being replaced by a generalized pressure tensor. Homogenous particle nucleation, transport, condensation and coagulation are described by a general dynamic equation, which is solved using the method of moments. The results show that the maximal nucleation rate and number density occurs near the nozzle exit, and particle precipitation inside the nozzle might not be ignored. Particles grow mainly across the shocks. Fluid in the shear layer of the jet shows a relatively low temperature, high nucleation rate, and carries particles with small sizes. On the plate, particles within the jet have smaller average size and higher geometric mean, while particles outside the jet shows a larger average size and a lower geometric mean. Increasing the preexpansion temperature will increase both the average particle size and standard deviation. The preexpansion pressure does not show a monotonic dependency with the average particle size. Increasing the distance between the plate and the nozzle exit might decrease the particle size. For all the cases in this paper, the average particle size on the plate is on the order of tens of nanometers.

Partial Regularity of Suitable Weak Solutions to the Fractional Navier–Stokes Equations

In this paper, we study the partial regularity of fractional Navier–Stokes equations in \({\mathbb{R}^3 \times (0, \infty)}\) with \({3/4 < s < 1}\) . We show that the suitable weak solution is regular away from a relatively closed singular set whose (5−4s)-dimentional Hausdorff measure is zero. The result is a generalization of the partial regularity for the classical Navier–Stokes system in Caffarelli et al. (Commun Pure Appl Math 35:771–831, 1982).

Heat Transfer Problem in a Van Der Waals Gas

This paper is devoted to the study of a heat transfer problem in a van der Waals gas. We found that the extended thermodynamics theory for such a fluid predicts the non-vanishing dynamic pressure and the shear stress tensor, even in the simplest stationary planar case. However, the order of magnitude of the non-equilibrium variables is very small and, consequently, their experimental observation seems to be quite difficult. A comparison between the results derived from the classical Navier---Stokes Fourier theory and those from extended thermodynamics is also presented.

A Study of Linear Waves Based on Extended Thermodynamics for Rarefied Polyatomic Gases

We study the dispersion relation for sound in rarefied polyatomic gases basing on the recently developed theory of extended thermodynamics (ET) for both dense and rarefied polyatomic gases. For hydrogen and deuterium gases in a wide temperature range where the rotational and vibrational modes in a molecule play a role, we compare the dispersion relations with those obtained in experiments and by the classical Navier---Stokes Fourier theory. From the comparison with experiments, we estimate the bulk viscosity and evaluate its temperature dependence. We study the characteristics of attenuation in a gas which has a larger relaxation time related to the dynamic pressure than the other relaxation times related to the shear stress and the heat flux by adopting the ET theory with 6 fields.

On the Effective Viscosity Expression for Modeling Squeeze-Film Damping at Low Pressure

While at high pressure, the classical Navier–Stokes equation is suitable for modeling squeeze-film damping, at low pressure, it needs some modification in order to consider fluid rarefaction. According to a common approach, fluid rarefaction can be included in this equation by substituting the standard fluid viscosity with a fictitious quantity, known as effective viscosity, for which different formulations were proposed. In order to identify which expression works better, the results obtained when either formulation is implemented inside the Navier–Stokes equation (that is then solved by both analytical and numerical means) are compared with already available experimental data. At the end, a novel expression is discussed, derived from a computer-assessed optimization procedure.

Global existence of strong solutions to micropolar equations in cylindrical domains

The micropolar equations are a useful generalization of the classical Navier–Stokes model for fluids with microstructure. We prove the existence of global and strong solutions to these equations in cylindrical domains in . We do not impose any restrictions on the magnitude of the initial and external data, but we require that they cannot change in the x3‐direction too fast. Copyright © 2014 John Wiley &amp; Sons, Ltd.

The bond-based peridynamic system with Dirichlet-type volume constraint

In this paper, the bond-based peridynamic system is analysed as a non-local boundary-value problem with volume constraint. The study extends earlier works in the literature on non-local diffusion and non-local peridynamic models, to include non-positive definite kernels. We prove the well-posedness of both linear and nonlinear variational problems with volume constraints. The analysis is based on some non-local Poincaré-type inequalities and the compactness of the associated non-local operators. It also offers careful characterizations of the associated solution spaces, such as compact embedding, separability and completeness. In the limit of vanishing non-locality, the convergence of the peridynamic system to the classical Navier equations of elasticity with Poisson ratio ¼ is demonstrated.

Local in time results for local and non-local capillary Navier–Stokes systems with large data

In this article we study three capillary compressible models (the classical local Navier–Stokes–Korteweg system and two non-local models) for large initial data, bounded away from zero, and with a reference pressure state ρ¯ which is not necessarily stable (P′(ρ¯) can be non-positive). We prove that these systems have a unique local in time solution and we study the convergence rate of the solutions of the non-local models towards the local Korteweg model. The results are given for constant viscous coefficients and we explain how to extend them for density dependant coefficients.

The motion of a piezoviscous fluid under a surface load

Using a variant of a spectral collocation method we numerically solve the problem of the motion of a highly viscous fluid with pressure dependent viscosity under a surface load, which is a problem relevant in many applications, in particular in geophysics and polymer melts processing. We compare the results with the results obtained by the classical Navier–Stokes fluid (constant viscosity). It turns out that for a realistic parameter values the two models give substantially different predictions concerning the motion of the free surface and the velocity and the pressure fields beneath the free surface.As a byproduct of the effort to test the numerical scheme we obtain an analytical solution—for the classical Navier–Stokes fluid—of the surface load problem in a layer of finite depth.

Unifying binary fluid diffuse-interface models in the sharp-interface limit

AbstractRecent results published by Gugenberger et al. on surface diffusion (Phys. Rev. E, vol. 78, 2008, 016703), show that the sharp-interface limit of the phase field models often adopted in the literature fails to produce the appropriate boundary conditions. With this knowledge, we consider the sharp-interface limit of phase field models for binary fluids, obtained carefully, where hydrodynamic equations are coupled to phase field evolution based on Cahn–Hilliard or Allen–Cahn theories, in a variety of guises, and unify and contrast their forms and behaviours in the sharp-interface limit. In particular, a tensorial mobility model is analysed, which allows the bulk fluids in the outer region to satisfy classical Navier–Stokes type equations to all orders in the Cahn number.

Elastic stability of functionally graded rectangular plates under mechanical and thermal loadings

In this paper, the buckling of a functionally graded plate is studied. The material properties of the plate are assumed to be graded continuously in the direction of thickness. The variation of the material properties follows a simple power-law distribution in terms of the volume fractions of constituents. The plates are subjected to be under three types of mechanical loadings, namely; uniaxial compression along the x-axis, uniaxial compression along the y-axis, and biaxial compression, two types of thermal loading, namely; uniform temperature rise and linear temperature rise. The equilibrium and stability equations are derived using the classical plate theory (Kirchhoff theory) and Navier's solution. Resulting equations are employed to obtain the closed-form solution for the critical buckling load for each loading case. The results are verified with the known data in the literature. Key words: Buckling, classical plate theory, functionally graded materials, mechanical and thermal loads.

Numerical predictions of backward-facing step flows in microchannels using extended Navier–Stokes equations

Flows in microchannels were successfully predicted, in the past, both analytically and numerically, employing the extended Navier–Stokes equations (ENSE). In ENSE, the self-diffusion transport of mass, together with the resulting momentum and heat transport, is taken into account properly and the same is omitted in the classical Navier–Stokes equations. The ENSE have been employed here to numerically predict backward-facing step flows in microchannels, and the predictions are summarized in this paper. The results obtained by employing ENSE are compared with the available literature data computed by both direct simulation Monte Carlo and slip-velocity-based simulations. The good agreement of the present results with those given in the literature evidently points out that the ENSE can be applied to gas flows through complex microchannel geometries.