Natural Convection in Shallow Enclosures With Differentially Heated Endwalls

1988 ◽  
Vol 110 (3) ◽  
pp. 625-634 ◽  
Author(s):  
S. Paolucci ◽  
D. R. Chenoweth
Keyword(s):  
Numerical Solutions ◽  
Stokes Equations ◽  
Ideal Gas ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
The Core ◽  
Nusselt Numbers ◽  
Aspect Ratios ◽  
Ideal Gas Law ◽  
Rayleigh Numbers

We consider a low-aspect-ratio two-dimensional rectangular cavity, differentially heated with an arbitrarily large horizontal temperature difference. Steady-state results obtained from numerical solutions of the transient Navier-Stokes equations are given for air using the ideal gas law and Sutherland law transport properties. We clarify the different flow regimes possible by using numerical results for aspect ratios 0.025 ≤ A ≤ 1 and for Rayleigh numbers (based on height) 102 ≤ Ra ≤ 109. We present Nusselt numbers, and temperature and velocity distributions, and compare them with existing data. At high Ra in the Boussinesq limit we show the existence of weak secondary and tertiary flows in the core of the cavity. In addition we present novel solutions in the non-Boussinesq regime. We find that the classical parallel flow solution, accurate in the core of the cavity in the Boussinesq limit, does not exist when variable properties are introduced. For higher Rayleigh numbers, we generalize the well-known analytical boundary layer solution of Gill. The non-Boussinesq solutions show greatly reduced static pressure levels and lower critical Rayleigh numbers.

Similarity solutions for viscous vortex cores

1992 ◽  
Vol 238 ◽  
pp. 487-507 ◽  
Author(s):  
Ernst W. Mayer ◽  
Kenneth G. Powell
Keyword(s):  
Numerical Solutions ◽  
Stokes Equations ◽  
Axial Flow ◽  
Similarity Solutions ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
The Core ◽  
Axial Pressure Gradient ◽  
Self Similar ◽  
Similar Solutions

Results are presented for a class of self-similar solutions of the steady, axisymmetric Navier–Stokes equations, representing the flows in slender (quasi-cylindrical) vortices. Effects of vortex strength, axial gradients and compressibility are studied. The presence of viscosity is shown to couple the parameters describing the core growth rate and the external flow field, and numerical solutions show that the presence of an axial pressure gradient has a strong effect on the axial flow in the core. For the viscous compressible vortex, near-zero densities and pressures and low temperatures are seen on the vortex axis as the strength of the vortex increases. Compressibility is also shown to have a significant influence upon the distribution of vorticity in the vortex core.

Axisymmetric compressible flow in a rotating cylinder with axial convection

1985 ◽  
Vol 154 ◽  
pp. 121-144 ◽  
Author(s):  
Marius Ungarish ◽  
Moshe Israeli
Keyword(s):  
Compressible Flow ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Nonlinear Effects ◽  
Ideal Gas ◽  
Navier Stokes ◽  
Linear Perturbation ◽  
Linear Extensions ◽  
Navier Stokes Equations ◽  
Thermally Conducting

The steady compressible flow of an ideal gas in a rotating annulus with thermally conducting walls is considered for small Rossby number ε and Ekman number E and moderate rotational Mach numbers M. Attention is focused on nonlinear effects which show up when σ and εM2 are not small (σ = ε/HE½, H is the dimensionless height of the container). These effects are not properly predicted by the classical linear perturbation analysis, and are treated here by quasi-linear extensions.The extra work required by these extensions is only the numerical solution of one ordinary differential equation for the pressure.Numerical solutions of the full Navier–Stokes equations in the nonlinear range are presented, and the validity of the present approach is confirmed.

Cavity flows driven by buoyancy and shear

1972 ◽  
Vol 51 (2) ◽  
pp. 221-231 ◽  
Author(s):  
K. Torrance ◽  
R. Davis ◽  
K. Eike ◽  
P. Gill ◽  
D. Gutman ◽  
...  
Keyword(s):  
Numerical Solutions ◽  
Stokes Equations ◽  
Fluid Motion ◽  
Temperature Fields ◽  
Navier Stokes ◽  
Cavity Flows ◽  
Navier Stokes Equations ◽  
Moving Wall ◽  
Aspect Ratios ◽  
Range Of Values

Fluid motion driven by the combined effects of a moving wall and natura convection is examined for rectangular cavities with heightlwidth ratios of ½, 1 and 2. The Reynolds number and Prandtl number are held fixed at Re = 100 and Pr = 1; the Grashof number is varied over the range of values Gr = 0, ±104, ±106. Flow and temperature fields obtained from numerical solutions of the Navier-Stokes equations reveal a marked influence of buoyancy for the larger aspect ratios when Gr = ±106 and the dominance of buoyancy for all aspect ratios when Gr = ± 106. Results are compared with earlier work where possible and some observations are offered on the convergence of the numerical solutions.

Laminar Compressible Flow Over a Stationary Disk in a Rotating Cylinder

1979 ◽  
Vol 101 (2) ◽  
pp. 173-180 ◽  
Author(s):  
M. Toren ◽  
A. Solan
Keyword(s):  
Angular Velocity ◽  
Compressible Flow ◽  
Close Agreement ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Meridional Circulation ◽  
Rotating Cylinder ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
The Core

The laminar isothermal compressible flow in a rotating cylinder with a stationary bottom is treated both numerically and by boundary layer matching. The numerical solution of the Navier-Stokes equations is based on a modified Cheng-Allen finite difference scheme. An approximate solution is obtained by matching boundary layers to an interior core. For sufficiently low Ekman numbers, the approximate and numerical solutions are in close agreement. The compressibility is shown to increase the angular velocity in the core and to decrease the meridional circulation. When the aspect ratio of the cylinder is increased, both the angular velocity in the core and the meridional circulation increase.

Rayleigh-Be´nard Convection in Open and Closed Rotating Cavities

2005 ◽  
Author(s):  
Martin P. King ◽  
Michael Wilson ◽  
J. Michael Owen
Keyword(s):  
Stokes Equations ◽  
Navier Stokes ◽  
Turbine Engines ◽  
Gas Turbine Engines ◽  
Navier Stokes Equations ◽  
Nusselt Numbers ◽  
Published Evidence ◽  
Cooling Air ◽  
Rayleigh Numbers ◽  
Good Agreement

Buoyancy effects can be significant in the rotating annular cavities found between compressor discs in gas-turbine engines, where Rayleigh numbers above 1012 are common. In some engines, the cavity is ‘closed’, so that the air is confined between four rotating surfaces: two discs and inner and outer cylinders. In most engines, however, the cavity is ‘open’, and there is an axial throughflow of cooling air at the centre. For open rotating cavities, a review of the published evidence suggests a Rayleigh-Be´nard type of flow structure, in which, at the larger radii, there are pairs of cyclonic and anti-cyclonic vortices. The toroidal circulation created by the axial throughflow is usually restricted to the smaller radii in the cavity. For a closed rotating annulus, solution of the unsteady Navier-Stokes equations, for Rayleigh numbers up to 109, show Rayleigh-Be´nard convection similar to that found in stationary enclosures. The computed streamlines in the r-θ plane show pairs of cyclonic and anti-cyclonic vortices; but, at the larger Rayleigh numbers, the computed isotherms suggest that the flow in the annulus is thermally mixed. At the higher Rayleigh numbers, the computed instantaneous Nusselt numbers are unsteady and tend to oscillate with time. The computed time-average Nusselt numbers are in good agreement with the correlations for Rayleigh-Be´nard convection in a stationary enclosure, but they are significantly higher than the published empirical correlations for a closed rotating annulus.

Rayleigh-Bénard Convection in Open and Closed Rotating Cavities

2005 ◽  
Vol 129 (2) ◽  
pp. 305-311 ◽  
Author(s):  
Martin P. King ◽  
Michael Wilson ◽  
J. Michael Owen
Keyword(s):  
Stokes Equations ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Nusselt Numbers ◽  
Bénard Convection ◽  
Published Evidence ◽  
Benard Convection ◽  
Rayleigh Benard Convection ◽  
Rayleigh Numbers ◽  
Rayleigh Benard

Buoyancy effects can be significant in the rotating annular cavities found between compressor discs in gas-turbine engines, where Rayleigh numbers above 1012 are common. In some engines, the cavity is “closed” so that the air is confined between four rotating surfaces: two discs and inner and outer cylinders. In most engines, however, the cavity is “open” and there is an axial throughflow of cooling air at the center. For open rotating cavities, a review of the published evidence suggests a Rayleigh–Bénard type of flow structure, in which, at the larger radii, there are pairs of cyclonic and anti-cyclonic vortices. The toroidal circulation created by the axial throughflow is usually restricted to the smaller radii in the cavity. For a closed rotating annulus, solution of the unsteady Navier–Stokes equations, for Rayleigh numbers up to 109, show Rayleigh–Bénard convection similar to that found in stationary enclosures. The computed streamlines in the r-θ plane show pairs of cyclonic and anti-cyclonic vortices; but, at the larger Rayleigh numbers, the computed isotherms suggest that the flow in the annulus is thermally mixed. At the higher Rayleigh numbers, the computed instantaneous Nusselt numbers are unsteady and tend to oscillate with time. The computed time-averaged Nusselt numbers are in good agreement with the correlations for Rayleigh–Bénard convection in a stationary enclosure, but they are significantly higher than the published empirical correlations for a closed rotating annulus.

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.

NUMERICAL SOLUTIONS OF 2D AND 3D SLAMMING PROBLEMS

2021 ◽  
Vol 153 (A2) ◽  
Author(s):  
Q Yang ◽  
W Qiu
Keyword(s):  
Free Surface ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Type Equation ◽  
The Body ◽  
Navier Stokes ◽  
Time Step ◽  
Navier Stokes Equations ◽  
Highly Nonlinear ◽  
2D And 3D

Slamming forces on 2D and 3D bodies have been computed based on a CIP method. The highly nonlinear water entry problem governed by the Navier-Stokes equations was solved by a CIP based finite difference method on a fixed Cartesian grid. In the computation, a compact upwind scheme was employed for the advection calculations and a pressure-based algorithm was applied to treat the multiple phases. The free surface and the body boundaries were captured using density functions. For the pressure calculation, a Poisson-type equation was solved at each time step by the conjugate gradient iterative method. Validation studies were carried out for 2D wedges with various deadrise angles ranging from 0 to 60 degrees at constant vertical velocity. In the cases of wedges with small deadrise angles, the compressibility of air between the bottom of the wedge and the free surface was modelled. Studies were also extended to 3D bodies, such as a sphere, a cylinder and a catamaran, entering calm water. Computed pressures, free surface elevations and hydrodynamic forces were compared with experimental data and the numerical solutions by other methods.

scholarly journals On the Importance of the Influence of both Velocity and Aspect Ratios on the Occurrence of Bifurcation Phenomena within a Two-Sided Lid-Driven Enclosure

2015 ◽  
Vol 55 ◽  
pp. 160-172
Author(s):  
Fayçal Hammami ◽  
Nader Ben Cheikh ◽  
Brahim Ben Beya
Keyword(s):  
Stokes Equations ◽  
Numerical Study ◽  
Numerical Results ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Left Wall ◽  
Driven Cavity ◽  
Aspect Ratios ◽  
The Stability ◽  
The Right

This paper deals with the numerical study of bifurcations in a two-sided lid driven cavity flow. The flow is generated by moving the upper wall to the right while moving the left wall downwards. Numerical simulations are performed by solving the unsteady two dimensional Navier-Stokes equations using the finite volume method and multigrid acceleration. In this problem, the ratio of the height to the width of the cavity are ranged from H/L = 0.25 to 1.5. The code for this cavity is presented using rectangular cavity with the grids 144 × 36, 144 × 72, 144 × 104, 144 × 136, 144 × 176 and 144 × 216. Numerous comparisons with the results available in the literature are given. Very good agreements are found between current numerical results and published numerical results. Various velocity ratios ranged in 0.01≤ α ≤ 0.99 at a fixed aspect ratios (A = 0.5, 0.75, 1.25 and 1.5) were considered. It is observed that the transition to the unsteady regime follows the classical scheme of a Hopf bifurcation. The stability analysis depending on the aspect ratio, velocity ratios α and the Reynolds number when transition phenomenon occurs is considered in this paper.

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