Flows In Cylindrical Coordinates Research Articles

Formulation of a Galerkin spectral element–Fourier method for three-dimensional incompressible flows in cylindrical geometries

A primitive-variable formulation for simulation of time-dependent incompressible flows in cylindrical coordinates is developed. Spectral elements are used to discretise the meridional semi-plane, coupled with Fourier expansions in azimuth. Unlike previous formulations where special distributions of nodal points have been used in the radial direction, the current work adopts standard Gauss–Lobatto–Legendre nodal-based expansions in both the radial and axial directions. Using a Galerkin projection of the symmetrised cylindrical Navier–Stokes equations, all geometric singularities are removed as a consequence of either the Fourier-mode dependence of axial boundary conditions or the shape of the weight function applied in the Galerkin projection. This observation implies that in a numerical implementation, geometrically singular terms can be naively treated by explicitly zeroing their contributions on the axis in integral expressions without recourse to special treatments such as l'Hopital's rule. Exponential convergence of the method both in the meridional semi-plane and in azimuth is demonstrated through application to a three-dimensional analytical solution of the Navier–Stokes equations in which flow crosses the axis.

Numerical study of laminar separation over an annular backstep

Numerical simulations of laminar flows with recirculation regions were performed. The vorticity-stream function representations of the Navier-Stokes equations for 2-D flow in Cartesian coordinates and for axisymmetric flow in cylindrical coordinates were solved numerically. The algorithm developed to solve these systems of coupled, nonlinear partial differential equations implements a variable step Crank-Nicolson scheme with Richardson's extrapolation to discretize the time integration. It implements a 4th-order collocation method to solve the linear elliptic problems at the core of the computations. An iterative method is applied to solve for the nonlinearities over each time step. This 4th-order finite element algorithm was verified by simulating two nonparallel flow exact solutions to the Navier-Stokes equations to within the accuracy specified by the user. Next, the laminar flow field over a planar backstep located below a free-surface was computed and compared with available experimental data. Subsequently, the method was applied to investigate the flow over an inner-radius annular backstep. Solutions for steady flows over the annular backstep for Re values from 100 to 400 were computed; the Re is based on the maximum velocity upstream of the step and the outer radius of the annulus.

Theory of magnetically insulated electron flows in coaxial pulsed power transmission lines

The Cartesian magnetically insulated transmission line (MITL) theory of Mendel et al. [Appl. Phys. 50, 3830 (1979); Phys. Fluids 26, 3628 (1983)] is extended to cylindrical coordinates. A set of equations that describe arbitrary electron flows in cylindrical coordinates is presented. These equations are used to derive a general theory for laminar magnetically insulated electron flows. The laminar theory allows one to specify the potentials, fields, and densities across a coaxial line undergoing explosive electron emission at the cathode. The theory is different from others available in cylindrical coordinates in that the canonical momentum and total energy for each electron may be nonzero across the electron sheath. A nonzero canonical momentum and total energy for the electrons in the sheath allows the model to produce one-dimensional flows that resemble flows from lines with impedance mismatches and perturbing structures. The laminar theory is used to derive two new self-consistent cylindrical flow solutions: (1) for a constant density profile and (2) for a quadratic density profile of the form ρ=ρc[(r2m−r2)/(r2m−r2c)]. This profile is of interest in that it is similar to profiles observed in a long MITL simulation [Appl. Phys. 50, 4996 (1979)]. The theoretical flows are compared to numerical results obtained with two-dimensional (2-D) electromagnetic particle-in-cell (PIC) codes.

Similarity solutions that give rise to hyperbolicity and change of type in steady flow of a viscoelastic fluid

Similarity solutions have proved to be a very useful tool for the study of flows of viscoelastic fluids since they allow us to check numerical computations against them. We compute here hyperbolic regions of the vorticity for an upper convected Maxwell model using the similarity solution of Phan-Thien for flow between rotating disks and using the similarity solution of Menon for an accelerated flow in cylindrical coordinates. We show that the extra tension becomes enormous at the edge of disks of ordinary rheometers under operating conditions.

'Coriolis resonance' within a rotating duct

An investigation of the unsteady disturbances of a fixed frequency within a radial duct rotating at a set speed is presented. The flow is assumed to be compressible, inviscid, and of a fluid which is a perfect gas. Equations are developed for the steady and the unsteady parts of the flow in cylindrical coordinates. The unsteady disturbances are expressed by Fourier decomposition in angular position, distance into the duct, and in time. It is found that a resonance is possible when the frequency of flow disturbances is twice the shaft-rotation frequency, considering only the radial and tangential disturbances and not the radial and circumferential disturbances. The particular point at which the resonance occurs indicates the occurrence is due to the Coriolis force, which is only present in the radial and tangential directions. It is noted that the Coriolis force can only be present in open-ended ducts, such as those found in centrifugal compressors.