Spin-up from rest of a compressible fluid in a rapidly rotating cylinder

1992 ◽  
Vol 237 ◽  
pp. 413-434 ◽  
Author(s):  
Jae Min Hyun ◽  
Jun Sang Park
Keyword(s):  
Mach Number ◽  
Compressible Fluid ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Axial Direction ◽  
Ekman Layer ◽  
Infinite Cylinder ◽  
Full Time ◽  
Viscous Effects ◽  
Initial State

Spin-up flows of a compressible gas in a finite, closed cylinder from an initial state of rest are studied, The flow is characterized by small reference Ekman numbers, and the peripheral Mach number is O(1). Comprehensive numerical solutions have been obtained for the full, time-dependent compressible Navier-Stokes equations. The details of the flow, temperature, and density evolution are described. In the early phase of spin-up, owing to the thermoacoustic disturbances caused by the compressible Rayleigh effect, the flows are oscillatory, and this oscillatory behaviour is pronounced at higher Mach numbers. The principal dynamical role of the Ekman layer is dominant over moderate times of orders of the homogeneous spin-up timescales. Owing to the density stratification in the radial direction, the Ekman layer is thicker in the central region of the interior. The interior azimuthal flows are mainly uniform in the axial direction. As the Mach number increases, the rate of spin-up in the interior becomes slower, and the propagating shear front is more diffusive. Explicit comparisons with the results for an infinite cylinder are made to ascertain the contributions of the endwall disks. In contrast to the usual incompressible spin-up from rest, the viscous effects are relatively more important for the case of a compressible fluid.

Flow Driven by a Finite, Shrouded Spinning Disk With Suction

1985 ◽  
Vol 52 (4) ◽  
pp. 766-770 ◽  
Author(s):  
J. M. Hyun
Keyword(s):  
Steady State ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Ekman Layer ◽  
Initial State ◽  
Disk Model ◽  
Stationary Disk ◽  
Spinning Disk ◽  
Real Flow ◽  
Finite Configuration

Numerical solutions are presented for the flow driven by a spinning disk which forms an endwall of a finite, closed cylinder. The effects of imposing a uniform suction (or blowing) through the spinning disk in finite configuration are investigated. The Reynolds number is large and the cylinder aspect ratio is 0(1). Finite-difference techniques are employed to integrate the time-dependent Navier-Stokes equations. The initial state is taken to be a uniform axial motion. Integration is performed until an approximate steady state is attained. When there is no suction, the infinite disk model is shown to provide a qualitatively representative approximation to the flow in the central core region. As a suction (blowing) is imposed, the core rotation rate in the case of finite configuration becomes smaller (larger) than that for the case of no suction, which is in disagreement with the predictions of the infinite disk model. These significant discrepancies point to a fundamental difficulty of the infinite disk model to adequately describe the real flow infinite geometry when there is a mass flux across the system boundary. Plots showing the meridional stream function at various times are constructed. Details of the flow structure in the approximate steady state are analyzed. When there is a suction, a strong Ekman layer is present on the spinning disk but the Ekman layer on the stationary disk fades. When there is a blowing, a strong Ekman layer forms on the stationary disk. It is shown that the dynamic effects influencing the character of the flow are confined to these Ekman layers.

Viscosity Effects on the Radiation Hydrodynamics of Horizontal Cylinders

1991 ◽  
Vol 113 (4) ◽  
pp. 334-343 ◽  
Author(s):  
R. W. Yeung ◽  
C.-F. Wu
Keyword(s):  
Numerical Solutions ◽  
Stokes Equations ◽  
Small Body ◽  
Low Frequency ◽  
Body Motion ◽  
Navier Stokes ◽  
Inviscid Fluid ◽  
Viscous Effects ◽  
Navier Stokes Equations ◽  
Horizontal Cylinders

The problem of a body oscillating in a viscous fluid with a free surface is examined. The Navier-Stokes equations and boundary conditions are linearized using the assumption of small body-motion to wavelength ratio. Generation and diffusion of vorticity, but not its convection, are accounted for. Rotational and irrotational Green functions for a divergent and a vorticity source are presented, with the effects of viscosity represented by a frequency Reynolds number Rσ = g2/νσ3. Numerical solutions for a pair of coupled integral equations are obtained for flows about a submerged cylinder, circular or square. Viscosity-modified added-mass and damping coefficients are developed as functions of frequency. It is found that as Rσ approaches infinity, inviscid-fluid results can be recovered. However, viscous effects are important in the low-frequency range, particularly when Rσ is smaller than O(104).

Direct computation of the sound generated by vortex pairing in an axisymmetric jet

1999 ◽  
Vol 383 ◽  
pp. 113-142 ◽  
Author(s):  
BRIAN E. MITCHELL ◽  
SANJIVA K. LELE ◽  
PARVIZ MOIN
Keyword(s):  
Mach Number ◽  
Stokes Equations ◽  
Near Field ◽  
Prediction Method ◽  
Axial Direction ◽  
Navier Stokes ◽  
Far Field ◽  
Source Terms ◽  
Vortex Pairing ◽  
Field Sound

The sound generated by vortex pairing in axisymmetric jets is determined by direct solution of the compressible Navier–Stokes equations on a computational grid that includes both the near field and a portion of the acoustic far field. At low Mach number, the far-field sound has distinct angles of extinction in the range of 60°–70° from the jet's downstream axis which can be understood by analogy to axisymmetric, compact quadrupoles. As the Mach number is increased, the far-field sound takes on a superdirective character with the dominant sound directed at shallow angles to the jet's downstream axis. The directly computed sound is compared to predictions obtained from Lighthill's equation and the Kirchhoff surface method. These predictions are in good agreement with the directly computed data. The Lighthill source terms have a large spatial distribution in the axial direction necessitating the introduction of a model to describe the source terms in the region downstream of the last vortex pairing. The axial non-compactness of the quadrupole sources must be adequately treated in the prediction method.

scholarly journals Numerical solutions for transonic flow

1949 ◽  
Vol 196 (1044) ◽  
pp. 32-36 ◽  
Keyword(s):  
Mach Number ◽  
Exact Solutions ◽  
Compressible Fluid ◽  
Transonic Flow ◽  
Numerical Solutions ◽  
Free Stream ◽  
Two Dimensional ◽  
Transverse Axis ◽  
Two Dimensional Flow ◽  
Free Stream Mach Number

In a previous paper (Cherry 1947), the author has established a family of exact solutions for steady two-dimensional flow of a compressible fluid past a cylinder; the final formulae are given in theorem 6, equations (5.17) to (5.21). These formulae have now been evaluated (taking γ = 1.405) for the value T 1 = 0.05, corresponding to a free-stream Mach number of 0.510, and the streamlines are shown in figure 1. The cylindrical obstacle has a thickness ratio 0.93, but is markedly different from an ellipse, being almost exactly circular over its up- and downstream quadrants. The Mach number a t the ends of its transverse axis is 1.39. The flow is everywhere regular, but a small increase in the free-stream Mach number would be critical; a shock-line would begin to appear near the points on the surface where the tangent is inclined at about 25 or 30° to the direction of the free-stream.

Asymptotic Preserving Error Estimates for Numerical Solutions of Compressible Navier--Stokes Equations in the Low Mach Number Regime

2018 ◽  
Vol 16 (1) ◽  
pp. 150-183 ◽  
Author(s):  
Eduard Feireisl ◽  
Mária Lukáčová-Medviďová ◽  
Šárka Nečasová ◽  
Antonín Novotný ◽  
Bangwei She
Keyword(s):  
Mach Number ◽  
Error Estimates ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Asymptotic Preserving ◽  
Navier Stokes Equations ◽  
Low Mach Number

Shocks generated in a confined gas due to rapid heat addition at the boundary. I. Weak shock waves

1984 ◽  
Vol 393 (1805) ◽  
pp. 309-329 ◽  
Keyword(s):  
Mach Number ◽  
Stokes Equations ◽  
Weak Shock ◽  
Initial Boundary ◽  
Heating Time ◽  
Navier Stokes ◽  
Conduction Time ◽  
Initial State ◽  
Navier Stokes Equations ◽  
Wall Spacing

An inert compressible gas, confined between infinite parallel planar walls, is in an equilibrium state initially. Subsequently energy is added at the boundary during a period that is short compared to the acoustic time of the slot t' a (the wall spacing divided by the equilibrium sound speed), but larger than the mean time between molecular collisions. Conductive heating of a thin layer of gas adjacent to the wall induces a gas motion arising from thermal expansion. The small local Mach number at the layer edge has the effect of a piston on the gas beyond. A linear acoustic wave field is then generated in a thicker layer adjacent to the walls. Eventually nonlinear accumulation effects occur on a timescale that is longer than the initial heating time but short compared with t' a . A weak shock then appears at some well defined distance from the boundary. If the heating rate at the wall is maintained over the longer timescale, then a high temperature zone of conductively heated expanding gas develops. The low Mach number edge speed of this layer acts like a contact surface in a shock tube and supports the evolution of the weak shock propagating further from the boundary. One-dimensional, unsteady solutions to the complete Navier-Stokes equations for an inert gas are obtained by using perturbation methods based on the asymptotic limit t' a / t' c → 0, where t' c , the conduction time of the region, is the ratio of the square of the wall spacing to the thermal diffusivity in the initial state. The shock strength is shown to be related directly to the duration of the initial boundary heating.

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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