Viscous Flow Equations Research Articles

Laminar flow past a sinusoidal cavity

The dynamic and thermal properties of a laminar flow past a sinusoidal cavity are studied. Velocity and temperature fields inside and above the cavity are numerically determined from the viscous flow equations. Results are compared with the experimental data obtained with a model placed in an incompressible and uniform flow. The good agreement with the dynamic field configuration allows the extension of these calculations to other sinusoidal wall geometries, to other main flow velocities and to cases with suction at the wall. Examples of streamlines and isotherms are given for these various situations.

Flow field calculation and its experimental verification for inertia stage of marine gas turbine air intake filter

A numerical method is presented in this paper for the solution of flow field in marine gas turbine air intake filtration channel. The flow field was successfully calculated by this method, and aerodynamic characteristics were obtained for various types of filtration channels. This work is expected to be of practical importance for the design of such filters. Upstream difference was adopted to discretize the non-conservative type N-S equation for two-dimensional, time-dependnet, viscous and incompressible flow, and the stability, convergence, accuracy and artificial viscosity of the resulting difference equation were examined. This equation can be used to calculate viscous flows with Reynolds number up to tens of thousands. Also presented in this paper is a calculation method for treating wall vortex at boundary inflection points. Careful studies show that calculation based on the difference equational and wall vortex treatment proposed here are in good agreement with experimental results.

A study of a finite-difference method for the one-dimensional viscous heat conductive gas flow equation. Part I: A priori estimates and stability

Article A study of a finite-difference method for the one-dimensional viscous heat conductive gas flow equation. Part I: A priori estimates and stability was published on January 1, 1987 in the journal Russian Journal of Numerical Analysis and Mathematical Modelling (volume 2, issue 3).

Some general properties of plane-parallel viscous flows

An expression is obtained for the integral of the equations of plane-parallel viscous incompressible flow expressing conservation of the Bernoulli function along a certain family of lines which, as the viscosity tends to zero, go over into the streamlines. These lines also determine the direction of transfer of the vorticity of the viscous flow.

Numerical analysis of viscous shear flows around a circular cylinder.

Numerical solutions of the Navier-Stokes equations for viscous shear flows past an infinitely long circular cylinder are obtained for Reynolds numbers ranging from Re=1 to Re=40. The shear parameters, ε's investigated for each Re are 0.01, 0.05, 0.1, 0.2, 0.5 and 1. The outer boundary condition of the free stream is replaced by adding to a linear shear flow Oseen's exact solution for the perturbation stream function that has been obtained in the questions of the uniform flow, and the stream function, Ψs, on the surface is decided by imposing a condition that the pressure is single-valued. Numerical solutions for the drag, lift, moment and pressure distributions are presented, and their dependence on the shear parameter are investigated. The most interesting result is that lift gives rise to minus even in this Reynolds number range. Lastly, flow patterns are shown for Re=20.

A Numerical Analysis of the Three-Dimensional Viscous Flow in a Transonic Compressor Rotor and Comparison With Experiment

The numerical analysis of highly loaded transonic compressors continues to be of considerable interest. Although much progress has been made with inviscid analyses, viscous effects can be very significant, especially those associated with shock–boundary layer interactions. While inviscid analyses have been enhanced by the interactive inclusion of blade surf ace boundary layer calculations, it may be better in the long term to develop efficient algorithms to solve the full three-dimensional Navier–Stokes equations. Indeed, it seems that many phenomena of key interest, like tip clearance flows, may only be accessible to a Navier–Stokes solver. The present paper describes a computer program developed for solving the three-dimensional viscous compressible flow equations in turbomachine geometries. The code is applied to the study of the flowfield in an axial-flow transonic compressor rotor with an attempt to resolve the tip clearance flow. The predicted flow is compared with laser anemometry measurements and good agreement is found.

A study of a finite-difference method for one-dimensional viscous heat-conducting gas flow equations. Part II: Error estimates and realization

A study of a finite-difference method for one-dimensional viscous heat-conducting gas flow equations. Part II: Error estimates and realization

Effects of Waves on the Boundary Layer of a Surface-Piercing Body

The boundary-value problem for the boundary layer of a surface-piercing body is formulated in a rigorous manner in which proper consideration is given to the viscous-fluid free-surface boundary conditions. Simplifications that are appropriate for small-amplitude waves are investigated. To this end, order-of-magnitude estimates are derived for the flow field in the neighborhood of the body-boundary-layer/free-surface juncture. It is shown that, for laminar flow, the parameter Ak/ϵ, where Ak is the wave steepness and ϵ is the nondimensional boundary-layer thickness, is important for characterizing the flow. In particular, for Ak/ϵ sufficiently large such that the free-surface boundary conditions have a significant influence a consistent formulation requires the solution of higher-order viscous-flow equations. For turbulent flow, these conclusions cannot be reached with the same degree of certainty. Numerical results are presented for the model problem of a combination Stokes-wave/flat plate. For this initial investigation, the usual thin-boundary-layer equations were solved using a three-dimensional implicit finite-difference method. The calculations are for laminar and turbulent flow and both demonstrate and quantify the influence of waves on boundary-layer development. Calculations were made using both the small-amplitude-wave and more approximate free-surface boundary conditions. Both the external-flow pressure gradients and the free-surface boundary conditions are shown to have a significant influence. The former influence penetrates to a depth of about half the wavelength and the latter is confined to a region very close to the free surface.

On the macroscopic description of slow viscous flow past a random array of spheres

We give rigorous derivation of Brinkman's equation as the effective equation for slow viscous flow in porous media with high porosity. The medium is composed of spherical obstacles distributed randomly, and the microscopic flow is described by the Stokes equation. Along the road we giveW 12 convergence estimates for the point-sources approximation.

On the Fractal Dimension of Attractors for Viscous Incompressible Fluid Flows

We prove that attractors associated to various equations of viscous incompressible fluid flows have finite fractal dimension and lie in the set of $C^\infty $-functions. The first part of this article consists in an abstract formulation of our results; then in the second part, we apply these results to Navier–Stokes equations with nonhomogeneous boundary conditions to N.S.E. on a Riemannian manifold, to thermo-hydraulic equations and finally to magnetohydrodynamic equations.

Exit loss in viscous tube flow

Numerous errors abound in the perception and calculation of gas flow in short tubes. This paper will develop modified equations for short-tube viscous flow in circular tubes with either a choked or a nonchoked exit. Several practical examples will be presented for vacuum component and system analysis. These equations and techniques are then combined with corresponding information from molecular flow studies to develop a new composite approach to gas flow in the transition range. The final result is a simplified approach to computer solutions of gas flow for orifices, short tubes, and long tubes—from molecular flow to atmospheric pressure.

Vapor condensation on an inclined plate within a porous medium

In thermal power generation the process of heating a surface by condensing vapor is widespread. The authors consider the problem of vapor condensation on an inclined plane surface located within a porous medium. The vapor condenses on the outer surface of a moving condensate film. The velocity profiles in the liquid film were calculated using Brinkman's filtration equation, which is a simple superposition of Darcy's law and the equation of viscous flow in a porous medium. The temperature distribution was measured over the height of the granular layer by using a movable bar carrying thermocouples placed at various points in the layer. The results obtained with regard to effective thermal conductivity agree qualitatively with data presented.

Reply by Authors to G. Degrez

References 1 Casier, F., Deconinck, H., and Hirsch, Ch., A Class of Bidiagonal Schemes for Solving the Euler Equations, AIAA Journal, Vol. 22, Nov. 1984, pp. 1556-1563. 2 MacCormack, R. W., The Effect of Viscosity in Hypervelocity Impact Cratering, AIAA Paper 69-354, April 1969. 3 MacCormack, R. W., A Numerical Method for Solving the Equations of Compressible Viscous Flow, AIAA Journal, Vol. 20, Sept. 1982, pp. 1275-1281.

An application of a boundary element method to natural convection

The direct boundary element method is applied to the numerical modelling of thermal fluid flow in a transient state. The Navier-Stokes equations are considered under the Boussinesq approximation and the viscous thermal flow equations are expressed in terms of stream function, vorticity, and temperature in two dimensions. Boundary integral equations are derived using logarithmic potential and time-dependent heat potential as fundamental solutions. Boundary unknowns are discretized by linear boundary elements and flow domains are divided into a series of triangular cells. Charged points are translated upstream in the numerical evaluation of convective terms. Unknown stream function, vorticity, and temperature are staggered in the computational scheme. Simple iteration is found to converge to the quasi steady-state flow. Boundary solutions for two-dimensional examples at a Reynolds number 100 and Grashoff number 10 7 are obtained.

The effects of curvature on the flow field in rapidly rotating gas centrifuges

The effects of curvature on the fluid dynamics of rapidly rotating gas centrifuges are studied. A governing system of a linear partial differential equation and boundary conditions is derived based on a linearization of the equations for viscous compressible flow. This system reduces to the Onsager pancake model if the effects of curvature are neglected. Approximations to the solutions of the governing equations with and without curvature terms are obtained via a finite-element method. Two examples are considered: first where the flow is driven by a thermal gradient at the wall of the centrifuge, and then for the flow being driven by the introduction and removal of mass through the ends of the centrifuge. Comparisons of the results obtained show that, especially for the second example, the inclusion of the terms due to curvature in the model can have an appreciable effect on the solution.

Computational models of liquid metal ion sources

This work defines a computational model of the motion of ions from a liquid metal ion source in the interelectrode region. An Algol 68 program has been written which calculates the Laplace potential in an axially symmetric region by a variable mesh finite difference method, for arbitrary shaped boundaries, and the results have been compared with those of the finite element method. The effect of space charge can be important for these sources due to the large current densities near the emitter and a method for determining these effects by an iterative solution of Poisson's equation is described. Finally, the liquid motion on the needle is considered, which has been shown to be described, for a wide variety of different liquid metals and needle sizes, by the linear equations of slow viscous flow.

Densification kinetics of chromium carbide and its alloys in hot pressing

A study was made of the densification kinetics in the hot pressing of chromium carbide and its alloys under pressures of 10.8–21.6 MPa at temperatures of 1350–1500°C for Cr3C2 and 900–1200°C for KKhN-15 and KKhNF-15 alloys in the isothermal holding time range from 300 to 1500 sec. It was found that increasing the duration of isothermal holding at all the temperatures and pressures investigated increased the relative densities of the materials from 0.65–0.7 at the lowest pressures and temperatures to 0.98–0.99 at the maximum values of these parameters. It is shown that the densification of chromium carbide and its alloys with nickel and nickel-phosphorus binders in hot pressing is described by a generalized equation of volume viscous flow of a porous solid; for Cr3C2 it is determined by nonlinear creep and for KKhN-15 and KKhNF-15 alloys by a linear flow of the solid particles of their carbide components in their liquid cementing phases. The energies of activation for creep are: 3.8 eV for chromium carbide, 1.8 eV for KKhN-15, and 1.5 eV for KKhNF-15.

On the effect of torsion on a helical pipe flow

An orthogonal coordinate system along a generic spatial curve has been introduced, and the Navier-Stokes equations for a steady incompressible viscous flow have been explicitly written in this frame of reference. As an application the flow in a helical pipe has been studied, and, formdii 5f curvature and torsion small compared with the radius of the pipe, the flow has been considered as a perturbed Poiseuille flow. The result is that for curvatures and torsions of the same order and for low Reynolds number the curvature induces on the flow a first-order effect on the parameter e =κa, where κ is the curvature and a the radius of the pipe, while the effect of the torsion on the flow is of the second order in E. This last result disagrees with those of Wang (1981), who, adopting a non-orthogonal coordinate system, found a first-order effect of torsion on the flow.

A Numerical Method for Solving the Equations of Compressible Viscous Flow

Although much progress has already been made In solving problems in aerodynamic design, many new developments are still needed before the equations for unsteady compressible viscous flow can be solved routinely. This paper describes one such development. A new method for solving these equations has been devised that 1) is second-order accurate in space and time, 2) is unconditionally stable, 3) preserves conservation form, 4) requires no block or scalar tridiagonal inversions, 5) is simple and straightforward to program (estimated 10% modification for the update of many existing programs), 6) is more efficient than present methods, and 7) should easily adapt to current and future computer architectures. Computational results for laminar and turbulent flows at Reynolds numbers from 3 x 10(exp 5) to 3 x 10(exp 7) and at CFL numbers as high as 10(exp 3) are compared with theory and experiment.

Isovector fields for problems in the mechanics of solids and fluids

Practical aspects of the application of isovector methods to equations of balance are summarized. The dimension reduction argument given in the appendix together with programs in the REDUCE-2 algebraic manipulation language reduce the computation time from man-years to a few mornings work at a computer terminal. Specific applications to commonly encountered problems are given. All isovector fields for the following systems are exhibited: steady incompressible viscous flows in 2 spatial dimensions; steady thermal boundary layer equations in 2 spatial dimensions; unsteady incompressible viscous flows in 3 spatial dimensions; and dynamic thermoelasticity in 3 spatial dimensions. These results give immediate information concerning the connectivity of the solution sets and similarity variables. In particular, it is shown that only solutions of the unsteady incompressible viscous flow equations with the same local continuity classes can be connected.