Navier-Stokes Equations For Incompressible Flow Research Articles

Exact solution for entrance region flow between parallel plates

A method of obtaining an exact solution of the complete Navier-Stokes equations for incompressible flow has been developed to study the development of fluid flow between parallel plates. An analytical solution has been developed in the form of a convergent infinite series and a numerical solution obtained for Reynolds numbers of 10 and 500. These results have been compared with data available elsewhere. The method developed here is expected to have general application

A numerical study of laminar separation bubbles using the Navier-Stokes equations

The flow in a two-dimensional laminar separation bubble is analyzed by means of finite-difference solutions to the Navier-Stokes equations for incompressible flow. The study was motivated by the need to analyze high-Reynolds-number flow fields having viscous regions in which the boundary-layer assumptions are questionable. The approach adopted in the present study is to analyze the flow in the immediate vicinity of the separation bubble using the Navier-Stokes equations. It is assumed that the resulting solutions can then be patched to the remainder of the flow field, which is analyzed using boundary-layer theory and inviscid-flow analysis. Some of the difficulties associated with patching the numerical solutions to the remainder of the flow field are discussed, and a suggestion for treating boundary conditions is made which would permit a separation bubble to be computed from the Navier-Stokes equations using boundary conditions from inviscid and boundary-layer solutions without accounting for interaction between individual flow regions. Numerical solutions are presented for separation bubbles having Reynolds numbers (based on momentum thickness) of the order of 50. In these numerical solutions, separation was found to occur without any evidence of the singular behaviour at separation found in solutions to the boundary-layer equations. The numerical solutions indicate that predictions of separation by boundary-layer theory are not reliable for this range of Reynolds number. The accuracy and validity of the numerical solutions are briefly examined. Included in this examination are comparisons between the Howarth solution of the boundary-layer equations for a linearly retarded freestream velocity and the corresponding numerical solutions of the Navier-Stokes equations for various Reynolds numbers.

Numerical integration of the three-dimensional Navier-Stokes equations for incompressible flow

A method of numerically integrating the Navier-Stokes equations for certain three-dimensional incompressible flows is described. The technique is presented through application to the particular problem of describing thermal convection in a rotating annulus. The equations, in cylindrical polar co-ordinate form, are integrated with respect to time by a marching process, together with the solving of a Poisson equation for the pressure. A suitable form of the finite difference equations gives a computationally-stable long-term integration with reasonably faithful representation of the spatial and temporal characteristics of the flow.Trigonometric interpolation techniques provide accurate (discretely exact) solutions to the Poisson equation. By using an auxiliary algorithm for rapid evaluation of trigonometric transforms, the proportion of computation needed to solve the Poisson equation can be reduced to less than 25% of the total time needed to’ advance one time step. Computing on a UNIVAC 1108 machine, the flow can be advanced one time-step in 2 sec for a 14 × 14 × 14 grid upward to 96 sec for a 60 × 34 × 34 grid.As an example of the method, some features of a solution for steady wave flow in annulus convection are presented. The resemblance of this flow to the classical Eady wave is noted.

Perturbation analysis of the Navier-Stokes equations in Lagrangian form with selected linear solutions

The Navier-Stokes equations for incompressible flow in their Lagrangian form are taken as a starting point. A perturbation technique is then used to obtain first- and second-order sets of equations, and the general procedure for solving the equations to any order is given. The first-order equations yield interesting two- and three-dimensional motions that have some of the properties of ‘stirring’ ‘eddies’ and ‘turbulence’; it is suggested that various problems in turbulent motion might possibly be re-examined by means of these equations.

Some Exact Solutions of the Navier-Stokes and the Hydromagnetic Equations

Some exact, closed-form solutions of the Navier-Stokes equations for incompressible flow and of the hydromagnetic equations for high-conductivity, incompressible flow are presented. They can be considered to be generalizations of Taylor's solutions. The solutions are two dimensional and cellular containing a single-space Fourier component; the spatial behavior is chosen in such a way that the nonlinear inertial term and the pressure term cancel one another, leaving a linear system to be solved. The time behavior of the solutions is quite general. The solutions to the hydromagnetic equations are such that the velocity and the magnetic fields are parallel and decoupled. The velocity behaves as it does in the purely mechanical case while the magnetic field simply decays in time; there is no source term for it in the present treatment.

A solution of the Navier-Stokes and energy equations illustrating the response of skin friction and temperature of an infinite plate thermometer to fluctuations in the stream velocity

An exact solution of the Navier-Stokes equations for incompressible flow is derived under the conditions: (i) the flow is two-dimensional and is bounded by an infinite, plane, porous wall; (ii) the flow is independent of the distance parallel to the wall; (iii) the component of velocity parallel to the wall at a large distance from it fluctuates in time about a constant mean; (iv) the component of velocity normal to the wall is constant. It is found that the skin-friction fluctuations illustrate Lighthill’s (1954) theory of the behaviour of boundary layers subject to fluctuating pressure gradients. The amplitude of the skin-friction fluctuations rises with frequency, while the phase lead of the skin-friction over the main-stream-velocity fluctuation rises from zero at zero frequency to 7r/4 at very high frequencies. The velocity profile in the boundary layer fluctuates, and under certain transient conditions resembles that of a separated boundary layer, that is, a boundary layer with reverse flow close to the wall. With viscous dissipation of kinetic energy taken into account, the corresponding exact solution of the energy equation for an incompressible fluid with constant physical properties is derived under a condition of zero heat transfer between the fluid and the wall—the so-called ‘thermometer’ or ‘kinetic temperature’ problem. Whereas the velocity field consists of a mean flow and a first-harmonic fluctuation, the temperature field contains additionally a second-harmonic fluctuation. It is found that the mean temperature of the wall rises with frequency, and is ultimately proportional to the square root of the frequency. The first-harmonic fluctuation of the wall temperature lags behind the main-stream-velocity fluctuation by an amount which rises from zero at zero frequency to 1/4n at high frequencies, while the phase lag of the second-harmonic rises from zero at zero frequency but drops again to zero at high frequencies. The amplitude of the first-harmonic fluctuation tends to zero at high frequencies, whereas the amplitude of the second-harmonic fluctuation tends to a non-zero limit. Thus the residual temperature fluctuation of the wall at high frequencies has a frequency which is twice that of the fluctuating stream.