Calculation of the steady flow through a curved tube using a new finite-difference method

1980 ◽  
Vol 99 (3) ◽  
pp. 449-467 ◽  
Author(s):  
S. C. R. Dennis
Keyword(s):  
Finite Difference ◽  
Steady Flow ◽  
Stokes Equations ◽  
Steady Motion ◽  
Circular Cross Section ◽  
Navier Stokes ◽  
Dean Number ◽  
Numerical Work ◽  
Curved Tube ◽  
Flow Through

A numerical method is described which is suitable for solving the equations governing the steady motion of a viscous fluid through a slightly curved tube of circular cross-section but which is also applicable to the solution of any problem governed by the steady two-dimensional Navier–Stokes equations in the plane polar co-ordinate system. The governing equations are approximated by a scheme which yields finite-difference equations which are of second-order accuracy with respect to the grid sizes but which have associated matrices which are diagonally dominant. This makes them generally more amenable to solution by iterative techniques than the approximations obtained using standard central differences, while preserving the same order of accuracy.The main object of the investigation is to obtain numerical results for the problem of steady flow through a curved tube which corroborate previous numerical work on this problem in view of a recent paper (Van Dyke 1978) which tends to cast doubt on the accuracy of previous calculations at moderately high values of the Dean number; this is the appropriate Reynolds-number parameter in this problem. The present calculations tend to verify the accuracy of previous results for Dean numbers up to 5000, beyond which it is difficult to obtain accurate results. Calculated properties of the flow are compared with those obtained in previous numerical work, with the predictions of boundary-layer theory for large Dean numbers and with the predictions of Van Dyke (1978).

scholarly journals Quantifying Compressibility and Slip in Multiparticle Collision (MPC) Flow Through a Local Constriction

2021 ◽  
Author(s):  
Tahmina Akhter ◽  
Katrin Rohlf
Keyword(s):  
Stokes Equations ◽  
Equations Of Motion ◽  
Wall Slip ◽  
Reynolds Numbers ◽  
Ideal Gas ◽  
Navier Stokes ◽  
Numerical Work ◽  
Navier Stokes Equations ◽  
Flow Through ◽  
Generalized Boundary Condition

The flow of a compressible fluid with slip through a cylinder with an asymmetric local constriction has been considered both numerically, as well as analytically. For the numerical work, a particle-based method whose dynamics is governed by the multiparticle collision (MPC) rule has been used together with a generalized boundary condition that allows for slip at the wall. Since it is well known that an MPC system corresponds to an ideal gas and behaves like a compressible, viscous flow on average, an approximate analytical solution has been derived from the compressible Navier–Stokes equations of motion coupled to an ideal gas equation of state using the Karman–Pohlhausen method. The constriction is assumed to have a polynomial form, and the location of maximum constriction is varied throughout the constricted portion of the cylinder. Results for centerline densities and centerline velocities have been compared for various Reynolds numbers, Mach numbers, wall slip values and flow geometries.

Steady two-dimensional flow through a row of normal flat plates

1990 ◽  
Vol 210 ◽  
pp. 281-302 ◽  
Author(s):  
D. B. Ingham ◽  
T. Tang ◽  
B. R. Morton
Keyword(s):  
Finite Difference ◽  
Stokes Equations ◽  
Reynolds Numbers ◽  
Navier Stokes ◽  
Two Dimensional ◽  
Corner Singularities ◽  
Flat Plates ◽  
Navier Stokes Equations ◽  
Flow Through ◽  
Good Agreement

A numerical and experimental study is described for the two-dimensional steady flow through a uniform cascade of normal flat plates. The Navier–Stokes equations are written in terms of the stream function and vorticity and are solved using a second-order-accurate finite-difference scheme which is based on a modified procedure to preserve accuracy and iterative convergence at higher Reynolds numbers. The upstream and downstream boundary conditions are discussed and an asymptotic solution is employed both upstream and downstream. A frequently used method for dealing with corner singularities is shown to be inaccurate and a method for overcoming this problem is described. Numerical solutions have been obtained for blockage ratio of 50 % and Reynolds numbers in the range 0 [les ]R[les ] 500 and results for both the lengths of attached eddies and the drag coefficients are presented. The calculations indicate that the eddy length increases linearly withR, at least up toR= 500, and that the multiplicative constant is in very good agreement with the theoretical prediction of Smith (1985a), who considered a related problem. In the case ofR= 0 the Navier–Stokes equations are solved using the finite-difference scheme and a modification of the boundary-element method which treats the corner singularities. The solutions obtained by the two methods are compared and the results are shown to be in good agreement. An experimental investigation has been performed at small and moderate values of the Reynolds numbers and there is excellent agreement with the numerical results both for flow streamlines and eddy lengths.

scholarly journals Quantifying Compressibility and Slip in Multiparticle Collision (MPC) Flow Through a Local Constriction

2021 ◽  
Author(s):  
Tahmina Akhter ◽  
Katrin Rohlf
Keyword(s):  
Stokes Equations ◽  
Equations Of Motion ◽  
Wall Slip ◽  
Reynolds Numbers ◽  
Ideal Gas ◽  
Navier Stokes ◽  
Numerical Work ◽  
Navier Stokes Equations ◽  
Flow Through ◽  
Generalized Boundary Condition

The flow of a compressible fluid with slip through a cylinder with an asymmetric local constriction has been considered both numerically, as well as analytically. For the numerical work, a particle-based method whose dynamics is governed by the multiparticle collision (MPC) rule has been used together with a generalized boundary condition that allows for slip at the wall. Since it is well known that an MPC system corresponds to an ideal gas and behaves like a compressible, viscous flow on average, an approximate analytical solution has been derived from the compressible Navier–Stokes equations of motion coupled to an ideal gas equation of state using the Karman–Pohlhausen method. The constriction is assumed to have a polynomial form, and the location of maximum constriction is varied throughout the constricted portion of the cylinder. Results for centerline densities and centerline velocities have been compared for various Reynolds numbers, Mach numbers, wall slip values and flow geometries.

Bifurcation in steady laminar flow through curved tubes

1982 ◽  
Vol 119 ◽  
pp. 475-490 ◽  
Author(s):  
K. Nandakumar ◽  
Jacob H. Masliyah
Keyword(s):  
Stokes Equations ◽  
Outer Wall ◽  
Dual Solutions ◽  
Navier Stokes ◽  
Dean Number ◽  
Navier Stokes Equations ◽  
Vortex Solution ◽  
Duct Geometry ◽  
Flow Through ◽  
Curved Ducts

The occurrence of dual solutions in curved ducts is investigated through a numerical solution of the Navier-Stokes equations in a bipolar-toroidal co-ordinate system. With the shape of duct being the region formed by the natural co-ordinate surfaces, it was possible to alter the duct geometry gradually and preserve the prevailing form of the velocity field, in a manner suggested by Benjamin (1978).In addition to the Dean number Dn = Re/Rc½, a geometrical parameter that defines the shape of the duct was also varied systematically to study the bifurcation of a two-vortex solution into a two- and four-vortex solution. Dual solutions have been found for all geometrical shapes investigated here. Of particular interest are the shapes of a full circle and a semicircle with a curved outer wall.

scholarly journals Modified Reynolds Equation for Steady Flow Through a Curved Pipe

2021 ◽  
Vol 23 (2) ◽  
Author(s):  
A. Ghosh ◽  
V. A. Kozlov ◽  
S. A. Nazarov
Keyword(s):  
Steady Flow ◽  
Reynolds Equation ◽  
Stokes Equations ◽  
Asymptotic Methods ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Narrow Tube ◽  
Curved Pipe ◽  
Flow Through ◽  
Curvature And Torsion

AbstractA Reynolds equation governing the steady flow of a fluid through a curvilinear, narrow tube, with its derivation from Navier–Stokes equations through asymptotic methods is presented. The channel considered may have a rather large curvature and torsion. Approximations of the velocity and the pressure of the fluid inside the channel are constructed by artificially imposing appropriate boundary conditions at the inlet and the outlet. A justification for the approximations is provided along with a comparison with a simpler case.

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.

Numerical Study of Viscous Flows Inside Partially Filled Spinning and Coning Cylinders

1996 ◽  
Vol 118 (2) ◽  
pp. 335-340 ◽  
Author(s):  
Mohamed Selmi
Keyword(s):  
Stokes Equations ◽  
Numerical Study ◽  
Center Of Mass ◽  
Steady Motion ◽  
Practical Interest ◽  
Nonlinear Effects ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Practical Applications ◽  
Analysis And Design

This paper is concerned with the solution of the 3-D-Navier-Stokes equations describing the steady motion of a viscous fluid inside a partially filled spinning and coning cylinder. The cylinder contains either a single fluid of volume less than that of the cylinder or a central rod and a single fluid of combined volume (volume of the rod plus volume of the fluid) equal to that of the cylinder. The cylinder rotates about its axis at the spin rate ω and rotates about an axis that passes through its center of mass at the coning rate Ω. In practical applications, as in the analysis and design of liquid-filled projectiles, the parameter ε = τ sin θ, where τ = Ω/ω and θ is the angle between spin axis and coning axis, is small. As a result, linearization of the Navier-Stokes equations with this parameter is possible. Here, the full and linearized Navier-Stokes equations are solved by a spectral collocation method to investigate the nonlinear effects on the moments caused by the motion of the fluid inside the cylinder. In this regard, it has been found that nonlinear effects are negligible for τ ≈ 0.1, which is of practical interest to the design of liquid-filled projectiles, and the solution of the linearized Navier-Stokes equations is adequate for such a case. However, as τ increases, nonlinear effects increase, and become significant as ε surpasses about 0.1. In such a case, the nonlinear problem must be solved. Complete details on how to solve such a problem is presented.

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