Convective Instabilities in Fully Developed Flows

1968 ◽  
Vol 90 (1) ◽  
pp. 84-86 ◽  
Author(s):  
M. Sherman
Keyword(s):  
Rayleigh Number ◽  
Temperature Gradient ◽  
Body Force ◽  
Critical Rayleigh Number ◽  
Arbitrary Cross Section ◽  
The Body ◽  
Circular Channel ◽  
Square Channel ◽  
Force Direction ◽  
Boussinesq Fluid

This paper considers the possibility of inducing a convective secondary flow in the fully developed channel flow of a quasi-incompressible (Boussinesq) fluid. Instabilities of this type can only occur when the temperature gradient in the direction of the body force exceeds a certain critical value. This temperature gradient is proportional to the Rayleigh number of the fluid. We find that for channels of arbitrary cross section, the critical Rayleigh number is Rc ≥ 1360 (h/d)4 where h is the arbitrary channel’s maximum dimension in the body force direction and d is the diameter of an equal area circular channel. For two special geometries it is possible to improve the foregoing lower bound estimate to the critical Rayleigh number. In a circular channel Rc ≥ 3450 and in a square channel Rc ≥ 2480.

Lower Bounds to the Critical Rayleigh Number in Completely Confined Regions

1967 ◽  
Vol 34 (2) ◽  
pp. 308-312 ◽  
Author(s):  
M. Sherman ◽  
S. Ostrach
Keyword(s):  
Rayleigh Number ◽  
Lower Bounds ◽  
Body Force ◽  
Critical Rayleigh Number ◽  
Convective Motion ◽  
The Body ◽  
Maximum Dimension ◽  
Fixed Temperature ◽  
Boussinesq Fluid ◽  
The Given

A method is presented for estimating lower bounds to the minimum Rayleigh number that will induce a state of convective motion in a quasi-incompressible (Boussinesq) fluid where the temperature gradient is in the direction of the body force. The fluid is completely confined by fixed-temperature, rigid bounding walls. For any arbitrary region, the critical Rayleigh number is greater than 1558(h/D)4 where h is the maximum dimension of the given region in the direction of the body force and D is the diameter of an equal volume sphere. In certain simple geometrical configurations, improved lower-bound estimates are calculated.

Lower Bounds to Thermal Instability Criteria of Completely Confined Fluids Inside Cylinders of Arbitrary Cross Section

1964 ◽  
Vol 31 (3) ◽  
pp. 376-379 ◽  
Author(s):  
D. Pnueli
Keyword(s):  
Cross Section ◽  
Lower Bounds ◽  
Thermal Instability ◽  
Critical Rayleigh Number ◽  
Arbitrary Cross Section ◽  
The Body ◽  
Instability Criterion ◽  
Confined Fluids ◽  
Force Direction ◽  
Rayleigh Numbers

A method is developed to compute the lower bounds for the thermal instability criterion (the critical Rayleigh number) for fluids completely confined inside cylinders of arbitrary cross section; i.e., Rayleigh numbers below which no spontaneous flow may occur in spite of the density gradient being opposite to the body force direction.

Effect of a stabilizing gradient of solute on thermal convection

1968 ◽  
Vol 34 (2) ◽  
pp. 315-336 ◽  
Author(s):  
George Veronis
Keyword(s):  
Rayleigh Number ◽  
Temperature Gradient ◽  
Prandtl Number ◽  
Critical Rayleigh Number ◽  
Effective Diffusion ◽  
Stratified Fluid ◽  
Finite Amplitude ◽  
Oscillatory Motion ◽  
Linear Stability Theory ◽  
Onset Of Convection

A stabilizing gradient of solute inhibits the onset of convection in a fluid which is subjected to an adverse temperature gradient. Furthermore, the onset of instability may occur as an oscillatory motion because of the stabilizing effect of the solute. These results are obtained from linear stability theory which is reviewed briefly in the following paper before finite-amplitude results for two-dimensional flows are considered. It is found that a finite-amplitude instability may occur first for fluids with a Prandtl number somewhat smaller than unity. When the Prandtl number is equal to unity or greater, instability first sets in as an oscillatory motion which subsequently becomes unstable to disturbances which lead to steady, convecting cellular motions with larger heat flux. A solute Rayleigh number, Rs, is defined with the stabilizing solute gradient replacing the destabilizing temperature gradient in the thermal Rayleigh number. When Rs is large compared with the critical Rayleigh number of ordinary Bénard convection, the value of the Rayleigh number at which instability to finite-amplitude steady modes can set in approaches the value of Rs. Hence, asymptotically this type of instability is established when the fluid is marginally stratified. Also, as Rs → ∞ an effective diffusion coefficient, Kρ, is defined as the ratio of the vertical density flux to the density gradient evaluated at the boundary and it is found that κρ = √(κκs) where κ, κs are the diffusion coefficients for temperature and solute respectively. A study is made of the oscillatory behaviour of the fluid when the oscillations have finite amplitudes; the periods of the oscillations are found to increase with amplitude. The horizontally averaged density gradients change sign with height in the oscillating flows. Stably stratified fluid exists near the boundaries and unstably stratified fluid occupies the mid-regions for most of the oscillatory cycle. Thus the step-like behaviour of the density field which has been observed experimentally for time-dependent flows is encountered here numerically.

Finite-amplitude cellular convection in a fluidsaturated porous layer

1980 ◽  
Vol 373 (1753) ◽  
pp. 199-222 ◽  
Keyword(s):  
Rayleigh Number ◽  
Temperature Gradient ◽  
Heat Transport ◽  
Critical Rayleigh Number ◽  
Three Dimensional ◽  
Finite Amplitude ◽  
Onset Of Convection ◽  
Optimum Heat ◽  
Square Cells

The local nonlinear stability of thermal convection in fluid-saturated porous media, subjected to an adverse temperature gradient, is investigated. The critical Rayleigh number at the onset of convection and the corresponding heat transfer are determined. An approximate analytical method is presented to determine the form and amplitude of convection. To facilitate the determination of the physically preferred cell pattern, a detailed study of both two- and three-dimensional motions is made and a very good agreement with available experimental data is found. The finite-amplitude effects on the horizontal wavenumber, and the effect of the Prandtl number on the motion are discussed in detail. We find that, when the Rayleigh number is just greater than the critical value, two dimensional motion is more likely than three-dimensional motion, and the heat transport is shown to have two regions for n =1. In particular, it is shown that optimum heat transport occurs for a mixed horizontal plan form formed by the linear combination of general rectangular and square cells. Since an infinite number of steady-state finite-amplitude solutions exist for Rayleigh numbers greater than the critical number A c * , a relative stability criterion is discussed th at selects the realized solution as that having the maximum mean-square temperature gradient.

Experiments on the Onset of Longitudinal Vortices in Laminar Forced Convection Between Horizontal Plates

1971 ◽  
Vol 93 (4) ◽  
pp. 335-341 ◽  
Author(s):  
M. Akiyama ◽  
G. J. Hwang ◽  
K. C. Cheng
Keyword(s):  
Rayleigh Number ◽  
Temperature Gradient ◽  
Forced Convection ◽  
Critical Rayleigh Number ◽  
Profile Measurement ◽  
Axial Temperature Gradient ◽  
Longitudinal Vortices ◽  
Axial Temperature ◽  
Laminar Forced Convection ◽  
Limiting Case

An experimental investigation is carried out to determine the onset of longitudinal columnar vortices due to buoyant forces for fully developed laminar forced convection between two infinite horizontal plates, each wall subjected to identical uniform axial temperature gradient but maintained at temperatures T1 and T2 (T1 > T2, T1 < T2, and T1 = T2) at lower and upper surfaces, respectively. The limiting case with vanishing axial temperature gradient and heating from below (T1 > T2) is known to have a critical Rayleigh number of 1708 and is used to check the accuracy of the testing apparatus. The onset of secondary flow is determined by a direct flow-visualization technique using cigarette smoke, and confirmed by a transverse temperature-profile measurement using a single thermocouple traverse. Experimental results for the critical Rayleigh number are compared with theory and the agreement is found to be good.

The Thermal Instability of Completely Confined Fluids Inside Some Particular Configurations

1963 ◽  
Vol 85 (4) ◽  
pp. 346-354 ◽  
Author(s):  
S. Ostrach ◽  
D. Pnueli
Keyword(s):  
Thermal Stability ◽  
Experimental Investigation ◽  
Rayleigh Number ◽  
Thermal Instability ◽  
Body Force ◽  
Critical Rayleigh Number ◽  
Upper Bounds ◽  
Instability Criterion ◽  
Confined Fluids ◽  
Thermal Stability Of

This paper deals with the thermal stability of completely confined fluids subject to a body force and a temperature gradient which are parallel and oriented in the same direction. It describes a method to obtain upper bounds to the instability criterion (the critical Rayleigh number) for piecewise cylindrical configurations, and demonstrates the use of this method treating some particular practical configurations. These upper bounds are shown to coincide with the critical Rayleigh number under some conditions. An account of experimental investigation of three of the particular configurations is presented and the experimental results compare favorably with the computed upper bounds.

Effect of Modulation on Thermal Convection Instability

2000 ◽  
Vol 55 (11-12) ◽  
pp. 957-966 ◽  
Author(s):  
P. K. Bhatia ◽  
B. S. Bhadauria
Keyword(s):  
Fourier Series ◽  
Rayleigh Number ◽  
Temperature Gradient ◽  
Stability Problem ◽  
Fluid Layer ◽  
Critical Rayleigh Number ◽  
Time Dependent ◽  
The Stability ◽  
Linear Stability Problem ◽  
Oscillatory Temperature

Abstract The linear stability problem for a fluid in a classic Benard configuration is considered. The applied temperature gradient is the sum of a steady component and a time-dependent periodic component. Only infinitesimal disturbances are considered. The time-dependent perturbation is expressed in Fourier series. The shift in critical Rayleigh number is calculated and the modulating effect of the oscillatory temperature gradient on the stability of the fluid layer is examined. Some comparison is made with known results.

Effect of Modulation on Rayleigh-Benard Convection-II

2003 ◽  
Vol 58 (2-3) ◽  
pp. 176-182
Author(s):  
B. S. Bhadauria
Keyword(s):  
Rayleigh Number ◽  
Temperature Gradient ◽  
Fluid Layer ◽  
Critical Rayleigh Number ◽  
Periodic Part ◽  
Prandtl Numbers ◽  
Rigid Walls ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard ◽  
Horizontal Fluid Layer

The linear stability of a horizontal fluid layer, confined between two rigid walls, heated from below and cooled from above is considered. The temperature gradient between the walls consists of a steady part and a periodic part that oscillates with time. Only infinitesimal disturbances are considered. Numerical results for the critical Rayleigh number are obtained for various Prandtl numbers and for various values of the frequency. Some comparisons with known results have also been made.

scholarly journals Computational Modeling of Vortex Generators for Turbomachinery

2002 ◽  
Author(s):  
R. V. Chima
Keyword(s):  
Computational Models ◽  
Body Force ◽  
Secondary Flows ◽  
The Body ◽  
Vortex Generators ◽  
Force Model ◽  
Suction Surface ◽  
Near Surface ◽  
Compressor Rotor ◽  
Surface Particle

In this work computational models were developed and used to investigate applications of vortex generators (VGs) to turbomachinery. The work was aimed at increasing the efficiency of compressor components designed for the NASA Ultra Efficient Engine Technology (UEET) program. Initial calculations were used to investigate the physical behavior of VGs. A parametric study of the effects of VG height was done using 3-D calculations of isolated VGs. A body force model was developed to simulate the effects of VGs without requiring complicated grids. The model was calibrated using 2-D calculations of the VG vanes and was validated using the 3-D results. Then three applications of VGs to a compressor rotor and stator were investigated: 1. The results of the 3-D calculations were used to simulate the use of small casing VGs used to generate rotor preswirl or counterswirl. Computed performance maps were used to evaluate the effects of VGs. 2. The body force model was used to simulate large partspan splitters on the casing ahead of the stator. Computed loss buckets showed the effects of the VGs. 3. The body force model was also used to investigate the use of tiny VGs on the stator suction surface for controlling secondary flows. Near-surface particle traces and exit loss profiles were used to evaluate the effects of the VGs.

Linear stability of rotating convection in an imposed shear flow

1997 ◽  
Vol 350 ◽  
pp. 271-293 ◽  
Author(s):  
PAUL MATTHEWS ◽  
STEPHEN COX
Keyword(s):  
Rayleigh Number ◽  
Shear Flow ◽  
Eigenvalue Problem ◽  
Linear Stability ◽  
Thermal Convection ◽  
Critical Rayleigh Number ◽  
Rotation Vector ◽  
Linear Stability Theory ◽  
Fixed Temperature ◽  
Differential Motion

In many geophysical and astrophysical contexts, thermal convection is influenced by both rotation and an underlying shear flow. The linear theory for thermal convection is presented, with attention restricted to a layer of fluid rotating about a horizontal axis, and plane Couette flow driven by differential motion of the horizontal boundaries.The eigenvalue problem to determine the critical Rayleigh number is solved numerically assuming rigid, fixed-temperature boundaries. The preferred orientation of the convection rolls is found, for different orientations of the rotation vector with respect to the shear flow. For moderate rates of shear and rotation, the preferred roll orientation depends only on their ratio, the Rossby number.It is well known that rotation alone acts to favour rolls aligned with the rotation vector, and to suppress rolls of other orientations. Similarly, in a shear flow, rolls parallel to the shear flow are preferred. However, it is found that when the rotation vector and shear flow are parallel, the two effects lead counter-intuitively (as in other, analogous convection problems) to a preference for oblique rolls, and a critical Rayleigh number below that for Rayleigh–Bénard convection.When the boundaries are poorly conducting, the eigenvalue problem is solved analytically by means of an asymptotic expansion in the aspect ratio of the rolls. The behaviour of the stability problem is found to be qualitatively similar to that for fixed-temperature boundaries.Fully nonlinear numerical simulations of the convection are also carried out. These are generally consistent with the linear stability theory, showing convection in the form of rolls near the onset of motion, with the appropriate orientation. More complicated states are found further from critical.

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