Numerical solution of steady and unsteady flow over a vibrating airfoil in a channel

The work deals with a numerical solution of 2D steady and unsteady inviscid incompressible flow over the profile NACA 0012 in a channel. The finite volume method (FVM) in a form of cell-centered explicit schemes at quadrilateral C-mesh is used. Governing system of equations is the system of incompressible Euler equations. The method of artificial compressibility and time dependent method is applied to steady computations. The small disturbance theory (SDT) applied to a numerical solution of flow over a rotated profile by a small angle only is mentioned. Brief introduction is given to the Arbitrary (Semi) Lagrangian-Eulerian (ALE) method used for unsteady computations. Some numerical results of unsteady flow over a vibrating profile achieved by both SDT and ALE method are presented. Unsteady flow is caused by prescribed oscillations of the profile (one degree of freedom) fixed in an elastic axis. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Retracted: Unsteady flow and heat transfer on a rotating disk in a porous medium

This article has been retracted. See retraction notice DOI: 10.1002/mma.850 . An unsteady flow and heat transfer in a porous medium of a viscous incompressible fluid over a rotating disk in an otherwise ambient fluid are studied. The unsteadiness in the flow field is caused by the angular velocity of the disk which varies with time. The new self‐similar solution of the Navier–Stokes and energy equations is obtained numerically. The solution obtained here is not only the solution of the Navier–Stokes equations, but also of the boundary layer equations. Also, for a simple scaling factor, it represents the solution of the flow and heat transfer in the forward stagnation‐point region of a rotating sphere or over a rotating cone. The asymptotic behaviour of the solution for a large porosity or for a large independent variable is also examined. The surface shear stresses in the radial and tangential directions and the surface heat transfer increase as the acceleration parameter increases. Also, the surface shear stress in the radial direction and the surface heat transfer decrease with increasing porosity, but the surface shear stress in the tangential direction increases. Copyright © 2006 John Wiley & Sons, Ltd.     

Flow over a profile in a channel with dynamical effects

The work deals with a numerical solution of 2D inviscid incompressible flow over the profile NACA 0012 in a channel. The finite volume method in a form of cell‐centered scheme at quadrilateral C‐mesh is used. Governing system of equations is the system of Euler equations. Numerical results are partially compared with experimental data. Steady state solutions of the flow as well unsteady flows caused by prescribed oscillation of the profile were computed. The method of artificial compressibility and the time dependent method are used for computation of the steady state solution. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The free-parameter solution of the convection equations in a vertical cylinder with a volume heat source

An exact solution of the free-convection equations is constructed in the Oberbeck–Boussinesq approximation, describing the flow of a viscous heat-conducting fluid in a vertical cylinder of large radius when heated by radiation. The initial problem is reduced to an operator equation with an extremely non-linear operator, satisfying Schauder's theorem in C[0,1]. An iteration procedure is proposed for determining the independent parameter, that occurs in the solution, which enables three different values to be obtained and, correspondingly, three classes of solution of the initial problem. The linear stability of all the solutions obtained is investigated and it is shown that, for chosen values of the problem parameters, the most dangerous one is the plane wave mode and two instability mechanisms are present. The flow structure and the type of instability depend considerably on the values of the free parameter.     

A Spectral method determination of the first critical Rayleigh number in a cylindrical container

The onset of laminar axisymmetric Rayleigh–Bénard convection is investigated analytically for fluid in a cylindrical container. All surfaces are considered to be solid and no-slip for the flow, whereas for the thermal boundary conditions both a perfectly conducting and an insulated side wall are considered. The governing Boussinesq equations are perturbed and an approximate solenoidal flow field and a temperature field are determined, using the assumption of separation of variables. Subsequently, a Chebysev–Galerkin spectral method is employed to reduce the equations to a system of first-order nonlinear ordinary differential equations. The approximate representation of the flow and temperature fields make it possible to perform the calculations analytically. The first critical Rayleigh number (Ra_cr) is then calculated using local stability analysis. The resulting value of Ra_cr compares favorably with previous numerical and experimental studies. The analytical solution presented here allows for deeper insights into the physics of this extensively studied problem to be identified.     

Arbitrary Motions of a Viscous Incompressible Liquid in a Finite Region

By using an entirely constructive analysis it is shown that there exist solutions of the viscous flow equations in which two of the Cartesian components of velocity are arbitrary and the third component can be constructed uniquely by the application of integrability conditions. The solutions are in general defined in a finite region of the fluid and can be matched onto a well-defined deterministic solution of the Navier–Stokes equations such that all of the relevant physical quantities are continuous at the interface.     

Flow due to an oscillatory free stream past a rotating disk

Summary In this paper, an exact solution of the Navier-Stokes' equations for the flow due to the harmonic oscillations of a free stream past a rotating disk is presented. The differential equations for this flow, valid for all amplitudes and frequencies of oscillations, are obtained. Analytical-numerical solutions are developed for low and high frequency of oscillations and enough numbers of terms are used in the calculation to give the solution a satisfactory accuracy over the entire flow region.

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Zusammenfassung In dieser Arbeit wird eine exakte Lösung der Navier-Stokes' schen Gleichungen für die durch die harmonischen Schwingungen einer freien Strömung an einer rotierenden Scheibe verursachte Strömung angegeben. Die Differentialgleichungen für die Strömung, gültig für alle Amplituden und Oszillations-frequenzen, werden erhalten. Analytisch-numeriche Lösungen werden für hohe und niedrige Oszillations-frequenzen entwickelt und eine genügende Anzahl von Gliedern wird in der Berechnung gebraucht, um der Lösung eine befriedigende Genauigkeit über den ganzen Strömungsbereich zu geben.

     

Exact solution of the Navier-Stokes equations for the oscillating flow in a duct of a cross-section of right-angled isosceles triangle

The Navier-Stokes equations have been solved in order to obtain an analytical solution of the fully developed laminar flow in a duct having a cross section of a right-angled, isosceles triangle. We obtained a solution for the case of oscillating pressure gradient flow. The pulsating flow is obtained by the superposition of the steady and oscillating pressure gradient solutions.     

Steady flow of incompressible fluid between two co-rotating disks

This article provides an analytical solution of the Navier–Stokes equations for the case of the steady flow of an incompressible fluid between two uniformly co-rotating disks. The solution is derived from the asymptotical evolution of unknown components of velocity and pressure in a radial direction – in contrast to the Briter–Pohlhausen analytical solution, which is supported by simplified Navier–Stokes equations. The obtained infinite system of ordinary differential equations forms recurrent relations from which unknown functions can be calculated successively. The first and second approximations of solution are solved analytically and the third and fourth approximations of solutions are solved numerically. The numerical example demonstrates agreements with results obtained by other authors using different methods.     

Viscous flow over a shrinking sheet with a second order slip flow model

In this paper, viscous flow over a shrinking sheet is solved analytically using a newly proposed second order slip flow model. The closed solution is an exact solution of the full governing Navier–Stokes equations. The solution has two branches in a certain range of the parameters. The effects of the two slip parameters and the mass suction parameter on the velocity distribution are presented graphically and discussed. For certain combinations of the slip parameters, the wall drag force can decrease with the increase of mass suction. These results clearly show that the second order slip flow model is necessary to predict the flow characteristics accurately.     

Global Weak Solutions for a Parabolic System Modeling a One-Dimensional Miscible Flow in Porous Media

We consider an initial boundary value problem for a nonlinear differential system of two equations. Such a system is formed by the equations of compressible miscible flow in a one-dimensional porous medium. No assumption about the mobility ratio is involved. Under some reasonable assumptions on the data, we prove the existence of a global weak solution. Our basic approach is the semi-Galerkin method. We use the technique of renormalized solutions for parabolic equations in the derivation ofa prioriestimates.     

Unsteady viscous flow over a rotating stretchable disk with deceleration

In this work, the laminar unsteady flow over a stretchable rotating disk with deceleration is investigated. The three dimensional Navier–Stokes (NS) equations are reduced into a similarity ordinary differential equation group, which is solved numerically using a shooting method. Mathematically, two solution branches are found for the similarity equations. The lower solution branch may not be physically feasible due to a negative velocity in the circumferential direction. For the physically feasible solution branch, namely the upper solution branch, the fluid behavior is greatly influenced by the disk stretching parameter and the unsteadiness parameter. With disk stretching, the disk can be friction free in both the radial and the circumferential directions, depending on the values of the controlling parameters. The results provide an exact solution to the whole unsteady NS equations with new nonlinear phenomena and multiple solution branches.     

Simulation of Separating Flows over a Backward-Facing Step

Simulation results are reported for plane two-dimensional viscous incompressible flow in a channel with an abrupt expansion. The mathematical model is provided by the quasi-hydrodynamic equations in the incompressible fluid approximation. The computations are carried out in a range of Reynolds numbers including both laminar and turbulent flow. As the Reynolds number increases, the solution bifurcates and the steady laminar flow changes to time-dependent flow. The computation results are consistent with known experimental data. Turbulence models were not used for large Reynolds number computations.     

The instability of the flow in a suddenly blocked pipe

This paper considers the decay of Poiseuille flow within a suddenlyblocked pipe. For small to moderate times the flow is shownto consist of an inviscid core flow coupled with a boundarylayer at the pipe wall. A small-time asymptotic solution isdeveloped and it is shown that this solution is valid for timesup to the point at which the boundary layer fills the wholepipe. A small-time composite solution is used to initiate anumerical marching procedure which overcomes the small-timesingularity that arises in the flow and so allows us to describethe ultimate decay of the flow within a blocked pipe. The stabilityof this flow is then considered using both a quasi-steady approximationand a transient-growth analysis based upon marching solutionsof the linearized Navier–Stokes equations. Our transientstability analysis predicts a critical Reynolds number, fortransition to turbulence, in the range 970 < Re < 1370.     

Continuum models of rarefied gas flows in problems of hypersonic aerothermodynamics

The limits of applicability of continuum flow models in the problem of the hypersonic rarefied gas flow over blunt bodies are determined by an asymptotic analysis of the Navier–Stokes equations, the numerical solution of the viscous shock layer equations and the numerical and asymptotic solution of the thin viscous shock layer equations for low Reynolds numbers. It is shown that the thin viscous shock layer model gives correct values of the skin friction coefficient and the heat transfer coefficient in the transitional to free-molecule flow regime. The asymptotic solutions, the numerical solutions obtained within the framework of different continuum models, and the results of a calculation by Direct Simulation Monte Carlo method are compared.     

Stagnation slip flow and heat transfer over a nonlinear stretching sheet

Stagnation slip flow and heat transfer characteristics of a viscous fluid over a nonlinear stretching surface has been investigated. The governing partial differential equations are transformed to nonlinear ordinary differential equations using similarity transformations. The analytical solution of the nonlinear system is obtained in series form using the very efficient homotopy analysis method (HAM). Convergence of the series solution is shown explicitly. Important features of flow and heat transfer characteristics are plotted and discussed. Comparison is made with existing numerical results when the stagnation‐point and slip effects are excluded. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

Homotopy analysis method for solving the MHD flow over a non-linear stretching sheet

The problem of the boundary layer flow of an incompressible viscous fluid over a non-linear stretching sheet is considered. Homotopy analysis method (HAM) is applied in order to obtain analytical solution of the governing nonlinear differential equations. The obtained results are finally compared through the illustrative graphs with the exact solution and an approximate method. The compression shows that the HAM is very capable, easy-to-use and applicable technique for solving differential equations with strong nonlinearity. Moreover, choosing a suitable value of none–zero auxiliary parameter as well as considering enough iteration would even lead us to the exact solution so HAM can be widely used in engineering too.     

On critical reynolds numbers in plane channels with a sudden expansion at the entry

A method for the numerical solution of fluid dynamics equations is proposed. The evolution of the structure of the laminar flow with increasing Reynolds number and its behavior at the critical Reynolds number Re_cris analyzed. Various flow modes at Re = Re_cr are discussed.     

Initial-boundary-value problem of hyperbolic equations for blood flow in a vessel

In this paper, we study an initial-boundary-value problem of a system of hyperbolic, partial-differential equations that models blood flow in a vessel. The one-spatial-dimensional model assumes that blood flow in the vessel is an incompressible, homogeneous, Newtonian fluid which has a small Womersley number. Boundary conditions with either the pressure or the flow rate at each end of the vessel are considered, and the existence of the global solution is obtained using a form of Glimm's finite-difference scheme.     

Steady incompressible flow around objects in general coordinates with a multigrid solution method

Steady incompressible flow around objects in general coordinates is investigated. First, an overview of the popular approaches to discretize incompressible flow problems in general coordinates is given. It has been chosen to solve the equations on a staggered grid with contravariant flux unknowns and pressure as primitive variables. A solution method multigrid is used, with a line smoother able to deal with stretched cells. For flow problems around objects solved with a single block discretization periodic boundary, conditions are prescribed and adaptations for the discretization and the multigrid method are given. Steady flow around a circular cylinder and around an ellipse are presented. © 1994 John Wiley & Sons, Inc.