Numerical Solutions of the Navier-Stokes Equations for Axisymmetric Flows

1968 ◽  
Vol 10 (5) ◽  
pp. 389-401 ◽  
Author(s):  
D. R. Strawbridge ◽  
G. T. J. Hooper
Keyword(s):  
Numerical Solutions ◽  
Stokes Equations ◽  
Axisymmetric Flow ◽  
Labyrinth Seal ◽  
Taylor Vortex ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Taylor Vortex Flow ◽  
Concentric Cylinders ◽  
Flow Through

A numerical method is presented for the solution of the time dependent Navier-Stokes equations for the axisymmetric flow of an incompressible viscous fluid. The method is applied to the problems of Taylor-vortex flow about an enclosed rotating cylinder and between infinite concentric cylinders, and to the analysis of the flow through a labyrinth seal. The torque calculations, which show favourable agreement with experiment, and the resulting flow patterns are presented graphically.

Dynamic Coefficients of Labyrinth Gas Seals: A Comparison of Experimental Results and Numerical Calculations

2000 ◽  
Author(s):  
K. Kwanka ◽  
J. Sobotzik ◽  
R. Nordmann
Keyword(s):  
Stokes Equations ◽  
Labyrinth Seal ◽  
Navier Stokes ◽  
Labyrinth Seals ◽  
Navier Stokes Equations ◽  
Rotating Speed ◽  
Dynamic Coefficients ◽  
Flow Through ◽  
Gas Seals ◽  
Good Agreement

Non-contacting labyrinth seals are still the most common constructive elements used to minimize leakage losses in turbomachinery between areas with high pressure and areas with low pressure. Unfortunately, the leakage flow through the labyrinth seal generates forces which can have a great impact on the dynamics of the turborotor. Particularly in cases of instability, the turbomachinery is restricted in its power or rotating speed because of violent self-excited vibrations of the rotor. The occurrence of self-excited rotor vibrations due to lateral forces must definitely be excluded. To consider the labyrinth forces in Finite-Element prediction, a set of preferably exact dynamic coefficients is required. Numerical approaches used to calculate the coefficients are based on Navier-Stokes equations. A comparison with experimental data is essential for a validation of the calculation. The experimental identification is difficult, because of the littleness of the forces to be measured in gas seals. Especially the non-conservative coefficients, cross-coupled stiffness and direct damping, show a good agreement in both magnitude and trend depending on the entry swirl of the seal.

New boundary conditions in the transition régime

1968 ◽  
Vol 2 (3) ◽  
pp. 293-310 ◽  
Author(s):  
Carlo Cercignani ◽  
Gino Tironi
Keyword(s):  
Boundary Conditions ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Slip Flow ◽  
Navier Stokes ◽  
Main Body ◽  
Parallel Plates ◽  
Navier Stokes Equations ◽  
Concentric Cylinders ◽  
Flow Heat

Starting from the Boltzmann equation, new boundary conditions are derived to be matched with the Navier—Stokes equations, that are supposed to hold in the main body of a gas. The idea upon which this method is based goes back to Maxwell and Langmuir. Since the distribution function is supposed to be completely determined by the Navier—Stokes equations, this new set of boundary conditions extends in some sense the validity of the macroscopic equations to the transition and free molecular régimes. In fact, it is shown that the free molecular and slip flow régimes are correctly described by this method; the latter is also supposed to give a reasonable approximation for the complete range of Knudsen numbers. The new procedure is applied to different problems such as plane Couette flow, plane and cylindrical Poiseuile flow, heat transfer between parallel plates and concentric cylinders. Results are obtained and compared with the exact numerical solutions for the above-mentioned problems.

Numerical Solutions of Viscous Flow through a Pipe Orifice at Low Reynolds Numbers

1968 ◽  
Vol 10 (2) ◽  
pp. 133-140 ◽  
Author(s):  
R. D. Mills
Keyword(s):  
Numerical Solutions ◽  
Stokes Equations ◽  
Viscous Incompressible Fluid ◽  
Reynolds Numbers ◽  
Navier Stokes ◽  
Axially Symmetric ◽  
Navier Stokes Equations ◽  
Low Reynolds Numbers ◽  
Flow Through ◽  
Good Agreement

Numerical solutions of the Navier-Stokes equations have been obtained in the low range of Reynolds numbers for steady, axially symmetric, viscous, incompressible fluid flow through an orifice in a circular pipe with a fixed orifice/pipe diameter ratio. Streamline patterns and vorticity contours are presented as functions of Reynolds number. The theoretically determined discharge coefficients are in good agreement with experimental results of Johansen (2).

scholarly journals Co-rotating Taylor–Couette flow enclosed by stationary disks

2013 ◽  
Vol 716 ◽  
Author(s):  
M. Heise ◽  
Ch. Hoffmann ◽  
Ch. Will ◽  
S. Altmeyer ◽  
J. Abshagen ◽  
...  
Keyword(s):  
Stokes Equations ◽  
Mode Selection ◽  
Axisymmetric Flow ◽  
Taylor Vortex ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Stable Regime ◽  
Taylor Couette ◽  
Bifurcation Behaviour ◽  
Flow States

AbstractWe report results of a combined numerical and experimental study on axisymmetric and non-axisymmetric flow states in a finite-length, co-rotating Taylor–Couette system in the Taylor vortex regime but also in the Rayleigh stable regime for moderate Reynolds numbers (${\leq }1000$). We found the dominant boundary-driven axisymmetric circulation to play a crucial role in the mode selection and the bifurcation behaviour in this flow. A sequence of partially hysteretic transitions to other axisymmetric multi-cell flow states is observed. Furthermore, we observed spiral states bifurcating via a supercritical Hopf bifurcation out of these multi-cell states which strongly determine the shape of the spiral. Finally, an excellent agreement between experimental and numerical results of the full Navier–Stokes equations is found.

Steady flow through a channel with a symmetrical constriction in the form of a step

1980 ◽  
Vol 372 (1750) ◽  
pp. 393-414 ◽  
Keyword(s):  
Asymptotic Theory ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Volumetric Flow Rate ◽  
Reynolds Numbers ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Uniform Grids ◽  
Flow Through ◽  
Good Agreement

Numerical solutions of the Navier-Stokes equations are given for the steady, two-dimensional, laminar flow of an incompressible fluid through a channel with a symmetric constriction in the form of a semi-infinite step change in width. The flow proceeds from a steady Poiseuille velocity distribution far enough upstream of the step in the wider part of the channel to a corresponding distribution downstream in the narrower part and is assumed to remain symmetrical about the centre line of the channel. The numerical scheme involves an accurate and efficient centred difference treatment developed by Dennis & Hudson (1978) and results are obtained for Reynolds numbers, based on half the volumetric flow rate, up to 2000. For a step that halves the width of the channel it is found that very fine uniform grids, with 80 intervals spaced across half of the wider channel upstream, are necessary for resolution of the solution for the flow downstream of the onset of the step. Slightly less refined grids are adequate to resolve the flow upstream. The calculated flow ahead of the step exhibits very good agreement with the asymptotic theory of Smith (1979 b)for Reynolds numbers greater than about 100; indeed, comparisons of the upstream separation position and of the wall vorticity nearby are believed to yield the best agreement between numerical and asymptotic solutions yet found. Downstream there is also qualitative agreement regarding separation and reattachment as the grid size is refined in the numerical results.

scholarly journals A Comparison of Simplified Two-dimensional Flow Models Exemplified by Water Flow in a Cavern

2017 ◽  
Vol 64 (3-4) ◽  
pp. 141-154
Author(s):  
Dzmitry Prybytak ◽  
Piotr Zima
Keyword(s):  
Water Flow ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Two Dimensional ◽  
Navier Stokes Equations ◽  
Flow Models ◽  
Two Dimensional Flow ◽  
Flow Through ◽  
Potential Flow Model

AbstractThe paper shows the results of a comparison of simplified models describing a two-dimensional water flow in the example of a water flow through a straight channel sector with a cavern. The following models were tested: the two-dimensional potential flow model, the Stokes model and the Navier-Stokes model. In order to solve the first two, the boundary element method was employed, whereas to solve the Navier-Stokes equations, the open-source code library OpenFOAM was applied. The results of numerical solutions were compared with the results of measurements carried out on a test stand in a hydraulic laboratory. The measurements were taken with an ADV probe (Acoustic Doppler Velocimeter). Finally, differences between the results obtained from the mathematical models and the results of laboratory measurements were analysed.

Internal laminar flow

2018 ◽  
Author(s):  
Marcel Escudier
Keyword(s):  
Poiseuille Flow ◽  
Stokes Equations ◽  
Circular Cross Section ◽  
Navier Stokes ◽  
Parallel Plates ◽  
Concentric Cylinder ◽  
Navier Stokes Equations ◽  
Concentric Cylinders ◽  
Constant Viscosity ◽  
Pressure Driven Flow

In this chapter it is shown that solutions to the Navier-Stokes equations can be derived for steady, fully developed flow of a constant-viscosity Newtonian fluid through a cylindrical duct. Such a flow is known as a Poiseuille flow. For a pipe of circular cross section, the term Hagen-Poiseuille flow is used. Solutions are also derived for shear-driven flow within the annular space between two concentric cylinders or in the space between two parallel plates when there is relative tangential movement between the wetted surfaces, termed Couette flows. The concepts of wetted perimeter and hydraulic diameter are introduced. It is shown how the viscometer equations result from the concentric-cylinder solutions. The pressure-driven flow of generalised Newtonian fluids is also discussed.

scholarly journals Transient Pressure-Driven Electroosmotic Flow through Elliptic Cross-Sectional Microchannels with Various Eccentricities

Computation ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 27
Author(s):  
Nattakarn Numpanviwat ◽  
Pearanat Chuchard
Keyword(s):  
Laplace Transform ◽  
Electroosmotic Flow ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Mathieu Functions ◽  
Navier Stokes Equations ◽  
Flow Rates ◽  
Elliptic Cross ◽  
Flow Through ◽  
Relative Errors

The semi-analytical solution for transient electroosmotic flow through elliptic cylindrical microchannels is derived from the Navier-Stokes equations using the Laplace transform. The electroosmotic force expressed by the linearized Poisson-Boltzmann equation is considered the external force in the Navier-Stokes equations. The velocity field solution is obtained in the form of the Mathieu and modified Mathieu functions and it is capable of describing the flow behavior in the system when the boundary condition is either constant or varied. The fluid velocity is calculated numerically using the inverse Laplace transform in order to describe the transient behavior. Moreover, the flow rates and the relative errors on the flow rates are presented to investigate the effect of eccentricity of the elliptic cross-section. The investigation shows that, when the area of the channel cross-sections is fixed, the relative errors are less than 1% if the eccentricity is not greater than 0.5. As a result, an elliptic channel with the eccentricity not greater than 0.5 can be assumed to be circular when the solution is written in the form of trigonometric functions in order to avoid the difficulty in computing the Mathieu and modified Mathieu functions.

A numerical study of the interaction between unsteay free-stream disturbances and localized variations in surface geometry

1989 ◽  
Vol 209 ◽  
pp. 285-308 ◽  
Author(s):  
R. J. Bodonyi ◽  
W. J. C. Welch ◽  
P. W. Duck ◽  
M. Tadjfar
Keyword(s):  
Numerical Solutions ◽  
Free Stream ◽  
Stokes Equations ◽  
Numerical Study ◽  
Navier Stokes ◽  
Surface Geometry ◽  
Navier Stokes Equations ◽  
S Waves ◽  
Tollmien Schlichting

A numerical study of the generation of Tollmien-Schlichting (T–S) waves due to the interaction between a small free-stream disturbance and a small localized variation of the surface geometry has been carried out using both finite–difference and spectral methods. The nonlinear steady flow is of the viscous–inviscid interactive type while the unsteady disturbed flow is assumed to be governed by the Navier–Stokes equations linearized about this flow. Numerical solutions illustrate the growth or decay of the T–S waves generated by the interaction between the free-stream disturbance and the surface distortion, depending on the value of the scaled Strouhal number. An important result of this receptivity problem is the numerical determination of the amplitude of the T–S waves.

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