A combined Lagrangian method for simulation of axisymmetric gas-particle vortex flows

A combined fully Lagrangian approach for meshless modeling of unsteady axisymmetric vortex flows of a gas-particle mixture with an incompressible carrier phase is developed. The approach proposed is based on the combination of a meshless vortex method for calculating axisymmetric flows of the carrier phase described by the Navier–Stokes (or Euler) equations and the full Lagrangian method for calculating the parameters of the dispersed phase. The combination of these methods reduces the problem of modeling the two phase flows to the solution of a high-order system of ordinary differential equations for the coordinates of toroidal vortex elements in the carrier phase and the particle trajectories, the velocity components, and the components of the Jacobian of transformation from the Eulerian to the Lagrangian variables in the dispersed phase. The application of the method is illustrated by modeling the behavior of an admixture of inertial Stokes particles with a small mass concentration in unsteady flows like solitary vortex rings in a viscous carrier phase and groups of vortex rings in an effectively inviscid carrier phase.     

On high‐resolution schemes for solving unsteady compressible two‐phase dilute viscous flows

A high‐resolution numerical scheme based on the MUSCL–Hancock approach is developed to solve unsteady compressible two‐phase dilute viscous flow. Numerical considerations for the development of the scheme are provided. Several solvers for the Godunov fluxes are tested and the results lead to the choice of an exact Riemann solver adapted for both gaseous and dispersed phases. The accuracy of the scheme is proven step by step through specific test cases. These simulations are for one‐phase viscous flows over a flat plate in subsonic and supersonic regimes, unsteady flows in a low‐pressure shock tube, two‐phase dilute viscous flows over a flat plate and, finally, two‐phase unsteady viscous flows in a shock tube. The results are compared with well‐established analytical and numerical solutions and very good agreement is achieved. Copyright © 1999 John Wiley & Sons, Ltd.     

Drift of large-scale hot thermals in stratified air flows

A difference method developed for the calculation of the three-dimensional unsteady flows of a viscous heat-conducting gas is used to investigate the influence of a horizontal wind, stratified with respect to height, on the motion of the eruption clouds of a volcano. It is shown that as the intensity of the air flow rises and its stratification increases the rate of ascent of the thermal decreases. Data are obtained for the features of the motion of the vortex rings formed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 173–176, November–December, 1984.     

Incompressible Viscous Fluid Flows in a Thin Spherical Shell

Linearized stability of incompressible viscous fluid flows in a thin spherical shell is studied by using the two-dimensional Navier–Stokes equations on a sphere. The stationary flow on the sphere has two singularities (a sink and a source) at the North and South poles of the sphere. We prove analytically for the linearized Navier–Stokes equations that the stationary flow is asymptotically stable. When the spherical layer is truncated between two symmetrical rings, we study eigenvalues of the linearized equations numerically by using power series solutions and show that the stationary flow remains asymptotically stable for all Reynolds numbers.      

UNSTEADY CONFINED VISCOUS FLOWS WITH OSCILLATING WALLS AND MULTIPLE SEPARATION REGIONS OVER A DOWNSTREAM-FACING STEP

This paper presents computational solutions for unsteady viscous flows in channels with a downstream-facing step, followed by an oscillating floor. These solutions of the unsteady Navier–Stokes equations are obtained with a time-integration method using artificial compressibility in a fixed computational domain, which is obtained via a time-dependent coordinate transformation from the fluid domain with moving boundaries. The computational method is first validated for steady viscous flows past a downstream-facing step by comparison with previous numerical solutions and experimental results. This method is then used to obtain solutions for unsteady viscous flows with multiple separation regions over a downstream-facing step with oscillating walls, for which there are no previously known solutions. Thus, the present results may be used as benchmark solutions for the unsteady viscous flows with multiple separation regions between fixed and oscillating walls.     

Combined Forced and Free Convective Flow in a Vertical Porous Channel: The Effects of Viscous Dissipation and Pressure Work

Buoyant flow is analysed for a vertical fluid saturated porous layer bounded by an isothermal plane and an isoflux plane in the case of a fully developed flow with a parallel velocity field. The effects of viscous dissipation and pressure work are taken into account in the framework of the Oberbeck–Boussinesq approximation scheme and of the Darcy flow model. Momentum and energy balances are combined in a dimensionless nonlinear ordinary differential equation solved numerically by a Runge–Kutta method. Both cases of upward pressure force (upward driven flows) and of downward pressure force (downward driven flows) are examined. The thermal behaviour for upward driven flows and downward driven flows is quite different. For upward driven flows, the combined effects of viscous dissipation and pressure work may produce a net cooling of the fluid even in the case of a positive heat input from the isoflux wall. For downward driven flows, viscous dissipation and pressure work yield a net heating of the fluid. A general reflection on the roles played by the effects of viscous dissipation and pressure work with respect to the Oberbeck–Boussinesq approximation is proposed.     

Reduction of viscous flows having orthogonal magnetic and velocity field distributions

Summary Certain steady magnetohydrodynamic flows of a viscous incompressible fluid, in which the magnetic field is everywhere orthogonal to the velocity field, are related to viscous compressible flows having zero magnetic field. Examples are given to illustrate this relationship. Other linked magnetic and non-magnetic flows can be found using similar processes.     

Vorticity in plane parallel, axially symmetrical or spherical flows of viscous fluid

For viscous (barotropic or incompressible) fluids it is shown that, if the vorticity and the viscous force are orthogonal, vortex lines are convected by a vector field which fits with the velocity field when viscosity vanishes (extension of Helmholtz theorem); it is also found that energy remains constant along the field lines of this vector field (extension of Bernoulli theorem).If, moreover, vorticity and velocity are orthogonal too, the magnitude of the vorticity then behaves as the density of a fluid which flows along streamsheets according to this very same vector field. These properties are mainly encountered for plane parallel flows, axially symmetrical flows, spherical flows, but also for some other miscellaneous flow geometries such as unidirectional or radial flows. The set of the former three flows can even be characterized by these properties; that enhances this set of important flow geometries, avails a general view on vorticity behavior, and explains the great simplicity of vorticity equations in these cases. Numerous examples and comments are given for illustrating.     

Viscous dissipation effect on heat transfer characteristics of rectangular microchannels under slip flow regime and H1 boundary conditions

Viscous dissipation effect on heat transfer characteristics of a rectangular microchannel is studied. Flow is governed by the Navier–Stokes equations with the slip flow and temperature jump boundary conditions. Integral transform technique is applied to derive the temperature distribution and Nusselt number. The velocity distribution is taken from literature. The solution method is verified for the case where viscous dissipation is neglected. It is found that, the viscous dissipation is negligible for gas flows in microchannels, since the contribution of this effect on Nu number is about 1%. However, this effect should be taken into account for much more viscous flows, such as liquid flows. Neglecting this effect for a flat microchannel with an aspect ratio of 0.1 for Br=0.04 underestimates the Nu number about 5%.     

Finite‐volume‐type VOF method on dynamically adaptive quadtree grids

This paper describes a finite‐volume volume‐of‐fluid (VOF) method for simulating viscous free surface flows on dynamically adaptive quadtree grids. The scheme is computationally efficient in that it provides relatively fine grid resolution at the gas–liquid interface and coarse grid density in regions where flow variable gradients are small. Special interpolations are used to ensure volume flux conservation where differently sized neighbour cells occur. The numerical model is validated for advection of dyed fluid in unidirectional and rotating flows, and for two‐dimensional viscous sloshing in a rectangular tank. Copyright © 2004 John Wiley & Sons, Ltd.     

An experimental study of viscous vortex rings

The work focuses on the problem of stability and viscous decay of single vortex rings. A tentative classification scheme is proposed for vortex rings which is based on extensive hot-wire measurements of velocity in the ring core and wake, and flow visualization, viz. laminar, wavy, turbulence-producing, and turbulent. Prediction of vortex ring type is shown to be possible, at least approximately, based on the vortex ring Reynolds number alone. Linear growth rates of ring diameter with time are observed for all types of vortex rings, with different growth rates occurring for laminar and turbulent vortex rings. Data on the viscous decay of vortex rings are used to provide experimental confirmation of the accuracy of Saffman's equation for the velocity of propagation of a vortex ring.The work reported herein is supported through a grant of the Natural Sciences and Engineering Research Council of Canada. Special thanks are due to CAPES (Brazil) for the award of a scholarship to the senior author.     

On Plane-Parallel Separated Flows of an Incompressible Fluid near Flat Boundaries

On the basis of Stokes separated flows, examples of separated flows described by the Navier-Stokes equations of a viscous incompressible fluid are constructed. These flows are represented by series convergent in a certain non-zero neighborhood of a flat contour immersed in the flow. In this neighborhood, the series have the same structure as those for the basic Stokes flows. Examples of the regions in which the series segments chosen give only a slight deviation from the numerical solutions of the Navier-Stokes equations are presented. The comparison between inviscid separated flows (without the no-slip condition on the contour) and viscous flows of the same structure (with the no-slip condition) shows that the viscosity does not play a decisive role in the formation of separation or the type of streamline approach to or departure from the contour.     

An entropy fixed cell‐centered Lagrangian scheme

On the basis of the work [P.‐H. Maire, R. Abgrall, J. Breil, J. Ovadia, SIAM J. Sci. Comput. 29 (2007), 1781–1824], we present an entropy fixed cell‐centered Lagrangian scheme for solving the Euler equations of compressible gas dynamics. The scheme uses the fully Lagrangian form of the gas dynamics equations, in which the primary variables are cell‐centered. And using the nodal solver, we obtain the nodal viscous‐velocity, viscous‐pressures, antidissipation velocity, and antidissipation pressures of each node. The final nodal velocity is computed as a weighted sum of viscous‐velocity and antidissipation velocity, so do nodal pressures, whereas these weights are calculated through the total entropy conservation for isentropic flows. Consequently, the constructed scheme is conservative in mass, momentum, and energy; preserves entropy for isentropic flows, and satisfies a local entropy inequality for nonisentropic flows. One‐ and two‐dimensional numerical examples are presented to demonstrate theoretical analysis and performance of the scheme in terms of accuracy and robustness.Copyright © 2013 John Wiley & Sons, Ltd.     

Hydrodynamic structure of viscous flows in rotating convergent-divergent channels

The results of the numerical calculations of isothermal viscous flows in convergent-divergent channels axially rotating at a constant angular velocity are presented. With reference to the example of the calculations performed at Re = 2000 the difference in the flowfields in a fixed channel and that rotating at an angular velocity Ω₀ = 10 s^?1 in analyzed. The results of the calculations of viscous flows in convergent-divergent channels at Re = 2000 and Ω₀ = 10 s^?1 are also presented.     

The complementary RANS equations for the simulation of viscous flows

A complementary set of Reynolds‐averaged Navier–Stokes (RANS) equations has been developed for steady incompressible, turbulent flows. The method is based on the Helmholtz decomposition of the velocity vector field into a viscous and a potential components. In the complementary RANS solver a potential solution coexists with a viscous solution with the purpose of contributing to a fastest decay of the viscous solution in the far field. The proposed complementary RANS equations have been validated for steady laminar and turbulent flows. The computational results show that the complementary RANS solver is able to produce less grid‐dependent solutions than a conventional RANS solver. Copyright © 2005 John Wiley & Sons, Ltd.     

Reduced Basis Method for Optimal Control of Unsteady Viscous Flows

In this article we discuss the reduced basis method (RBM) for optimal control of unsteady viscous flows. RBM is a reduction method in which one can achieve the versatility of the finite element method or another for that matter and gain significant reduction in the number of degrees of freedom. The essential idea in this method is to define a reduced order subspace spanned by few basis elements and then obtain the solution via a Galerkin projection. We present several ways to define this subspace. Feasibility of the approach is demonstrated on two boundary control problems in cavity and wall bounded channel flows. Control action is effected through boundary surface movement on part of the solid wall. Application of RBM to the control problems leads to finite dimensional optimal control problems which are solved using Newton's method. Through computational experiments we demonstrate the feasibility and applicability of the reduced basis method for control of unsteady viscous flows.     

Computation of unsteady transonic flow over a fighter wing using a zonal Navier–Stokes/full‐potential method

An improved hybrid method for computing unsteady compressible viscous flows is presented. This method divides the computational domain into two zones. In the inner zone, the Navier–Stokes equations are solved using a diagonal form of an alternating‐direction implicit (ADI) approximate factorisation procedure. In the outer zone, the unsteady full‐potential equation (FPE) is solved. The two zones are tightly coupled so that steady and unsteady flows may be efficiently solved. Characteristic‐based viscous/inviscid interface boundary conditions are employed to avoid spurious reflections at that interface. The resulting CPU times are about 60% of the full Navier–Stokes CPU times for unsteady flows in non‐vector processing machines. Applications of the method are presented for a F‐5 wing in steady and unsteady transonic flows. Steady surface pressures are in very good agreement with experimental data and are essentially identical to the full Navier–Stokes predictions. Density contours show that shocks cross the viscous/inviscid interface smoothly, so that the accuracy of full Navier–Stokes equations can be retained with significant savings in computational time. Copyright © 1999 John Wiley & Sons, Ltd.     

Some basic viscous flows in micropolar fluids

Summary This paper analyzes some basic viscous flows of micropolar fluids. The problems ofCouette andPoiseuille flows between two parallel plates and a rotating fluid with a free surface, are solved using the theory of micropolar fluids. The results are presented graphically and compared with the classical ones, and the differences are discussed.     

A two-parameter model of wave regimes for viscous liquid film flows

Wave regimes of viscous liquid film flows are considered when the viscosity coefficients vary in a wide range. An approximate model system of differential equations with two external governing parameters for the film layer thickness and the local flow rate is derived. The viscous dissipation of a film layer is taken into account in this system more accurately than in the well-known one-parameter Shkadov model. New properties of linear and nonlinear waves caused by the hydrodynamic instability of high-viscous liquid flows under gravity and surface tension are found.