Thermal Convection Bifurcation Within Isosceles Triangular Cavities

2007 ◽  
Author(s):  
El Hassan Ridouane ◽  
Antonio Campo
Keyword(s):  
Thermal Convection ◽  
Pitchfork Bifurcation ◽  
Critical Value ◽  
Mirror Plane ◽  
Maximum Height ◽  
Hysteresis Phenomenon ◽  
Conservation Equations ◽  
Symmetrical Solution ◽  
Steady Solutions ◽  
Volume Method

Laminar thermal convection of air confined to an isosceles triangular cavity heated from the base and symmetrically cooled from the upper inclined walls has been investigated numerically. The system of transient conservation equations, subject to the proper boundary conditions, along with the equation of state assuming the air behaves as a perfect gas are solved with the finite volume method. In the conservation equations, the second-order-accurate QUICK scheme was used for the discretization of the convective terms and the SIMPLE scheme for the pressure-velocity coupling. The maximum height-to-base aspect ratio A is fixed at 0.5, while the Grashof number extends from a low Gr = 103 to a high Gr = 106. The influence of Gr on the flow and temperature patterns is analyzed and discussed for two opposing scenarios, one corresponds to increasing Gr and the other corresponds to decreasing Gr. It is found that two steady-state solutions are possible, excluding their solution images through a vertical mirror plane. The symmetrical solution prevails for relatively low Grashof numbers. However, as the Gr is gradually increased, a transition occurs at a critical value of Gr. Above this critical value of Gr, an asymmetrical solution exhibiting a pitchfork bifurcation arises and eventually becomes steady. The existing ranges of these unsteady and steady solutions are reported for the two opposing scenarios. Also, issues related to the observed hysteresis phenomenon are discussed in detail.

Two-dimensional flow of a viscous fluid in a channel with porous walls

1991 ◽  
Vol 227 ◽  
pp. 1-33 ◽  
Author(s):  
Stephen M. Cox
Keyword(s):  
Finite Time ◽  
Stokes Equations ◽  
Pitchfork Bifurcation ◽  
Time Dependent ◽  
High Rate ◽  
Navier Stokes ◽  
Recent Theory ◽  
Navier Stokes Equations ◽  
Two Dimensional Flow ◽  
Steady Solutions

We consider the flow of a viscous incompressible fluid in a parallel-walled channel, driven by steady uniform suction through the porous channel walls. A similarity transformation reduces the Navier-Stokes equations to a single partial differential equation (PDE) for the stream function, with two-point boundary conditions. We discuss the bifurcations of the steady solutions first, and show how a pitchfork bifurcation is unfolded when a symmetry of the problem is broken.Then we describe time-dependent solutions of the governing PDE, which we calculate numerically. We analyse these unsteady solutions when there is a high rate of suction through one wall, and the other wall is impermeable: there is a limit cycle composed of an explosive phase of inviscid growth, and a slow viscous decay. The inviscid phase ‘almost’ has a finite-time singularity. We discuss whether solutions of the governing PDE, which are exact solutions of the Navier-Stokes equations, may develop mathematical singularities in a finite time.When the rates of suction at the two walls are equal so that the problem is symmetrical, there is an abrupt transition to chaos, a ‘homoclinic explosion’, in the time-dependent solutions as the Reynolds number is increased. We unfold this transition by perturbing the symmetry, and compare direct numerical integrations of the governing PDE with a recent theory for ‘Lorenz-like’ dynamical systems. The chaos is found to be very sensitive to symmetry breaking.

An Unstructured Reacting Flow Solver With Coupled Implicit Solution of the Species Conservation Equations

2008 ◽  
Author(s):  
Ankan Kumar ◽  
Sandip Mazumder
Keyword(s):  
Species Conservation ◽  
Domain Size ◽  
Reacting Flow ◽  
Computational Domain ◽  
Conservation Equations ◽  
Flow Solver ◽  
As Species ◽  
Minimum Residual ◽  
Volume Method ◽  
Coupled Solution

Many reacting flow applications mandate coupled solution of the species conservation equations. A low-memory coupled solver was developed to solve the species transport equations on an unstructured mesh with implicit spatial as well as species-to-species coupling. First, the computational domain was decomposed into sub-domains comprised of geometrically contiguous cells—a process termed internal domain decomposition (IDD). This was done using the binary spatial partitioning (BSP) algorithm. Following this step, for each sub-domain, the discretized equations were developed using the finite-volume method, written in block implicit form, and solved using an iterative solver based on Krylov sub-space iterations, i.e., the Generalized Minimum Residual (GMRES) solver. Overall (outer) iterations were then performed to treat explicitness at sub-domain interfaces and non-linearities in the governing equations. The solver is demonstrated for a laminar ethane-air flame calculation with five species and a single reaction step, and for a catalytic methane-air combustion case with 19 species and 22 reaction steps. It was found that the best performance is manifested for sub-domain size of about 1000 cells, the exact number depending on the problem at hand. The overall gain in computational efficiency was found to be a factor of 2–5 over the block Gauss-Seidel procedure.

On perturbations of Jeffery-Hamel flow

1988 ◽  
Vol 186 ◽  
pp. 559-581 ◽  
Author(s):  
W. H. H. Banks ◽  
P. G. Drazin ◽  
M. B. Zaturska
Keyword(s):  
Stokes Equations ◽  
Temporal Stability ◽  
Pitchfork Bifurcation ◽  
Critical Value ◽  
Navier Stokes ◽  
Two Dimensional ◽  
Nonlinear Perturbations ◽  
Navier Stokes Equations ◽  
Weakly Nonlinear ◽  
Rigid Plane

We examine various perturbations of Jeffery-Hamel flows, the exact solutions of the Navier-Stokes equations governing the steady two-dimensional motions of an incompressible viscous fluid from a line source at the intersection of two rigid plane walls. First a pitchfork bifurcation of the Jeffery-Hamel flows themselves is described by perturbation theory. This description is then used as a basis to investigate the spatial development of arbitrary small steady two-dimensional perturbations of a Jeffery-Hamel flow; both linear and weakly nonlinear perturbations are treated for plane and nearly plane walls. It is found that there is strong interaction of the disturbances up- and downstream if the angle between the planes exceeds a critical value 2α2, which depends on the value of the Reynolds number. Finally, the problem of linear temporal stability of Jeffery-Hamel flows is broached and again the importance of specifying conditions up- and downstream is revealed. All these results are used to interpret the development of flow along a channel with walls of small curvature. Fraenkel's (1962) approximation of channel flow locally by Jeffery-Hamel flows is supported with the added proviso that the angle between the two walls at each station is less than 2α2.

On the direct initiation of gaseous detonations by an energy source

1994 ◽  
Vol 277 ◽  
pp. 227-248 ◽  
Author(s):  
Longting He ◽  
Paul Clavin
Keyword(s):  
Energy Source ◽  
Critical Radius ◽  
Critical Value ◽  
Blast Waves ◽  
Initiation Energy ◽  
Point Energy ◽  
Gaseous Detonations ◽  
Curvature Effects ◽  
Direct Initiation ◽  
Steady Solutions

A new criterion for the direct initiation of cylindrical or spherical detonations by a localized energy source is presented. The analysis is based on nonlinear curvature effects on the detonation structure. These effects are first studied in a quasi-steady-state approximation valid for a characteristic timescale of evolution much larger than the reaction timescale. Analytical results for the square-wave model and numerical results for an Arrhenius law of the quasi-steady equations exhibit two branches of solutions with a C-shaped curve and a critical radius below which generalized Chapman–Jouguet (CJ) solutions cannot exist. For a sufficiently large activation energy this critical radius is much larger than the thickness of the planar CJ detonation front (typically 300 times larger at ordinary conditions) which is the only intrinsic lengthscale in the problem. Then, the initiation of gaseous detonations by an ideal point energy source is investigated in cylindrical and spherical geometries for a one-step irreversible reaction. Direct numerical simulations show that the upper branch of quasi-steady solutions acts as an attractor of the unsteady blast waves originating from the energy source. The critical source energy, which is associated with the critical point of the quasi-steady solutions, corresponds approximately to the boundary of the basin of attraction. For initiation energy smaller than the critical value, the detonation initiation fails, the strong detonation which is initially formed decays to a weak shock wave. A successful initiation of the detonation requires a larger energy source. Transient phenomena which are associated with the intrinsic instability of the quasi-steady detonations branch develop in the induction timescale and may induce additional mechanisms close to the critical condition. In conditions of stable or weakly unstable planar detonations, these unsteady phenomena are important only in the vicinity of the critical conditions. The criterion of initiation derived in this paper works to a good approximation and exhibits the huge numerical factor, 106–108, which has been experimentally observed in the critical value of the initiation energy.

Diesel Electro-injector: A Numerical Simulation Code

1995 ◽  
Vol 117 (4) ◽  
pp. 792-798 ◽  
Author(s):  
P. Digesu ◽  
D. Laforgia
Keyword(s):  
Flow Rate ◽  
Parametric Analysis ◽  
Magnetic Force ◽  
Simulation Code ◽  
Conservation Equations ◽  
One Dimensional ◽  
Element Simulation ◽  
System Energy ◽  
The One ◽  
Volume Method

A simulation code of an electro-injector for diesel engines is presented with the preliminary parametric analysis carried out with the code. The simulation code is based upon the concentrated volume method as for the chambers of the system. Energy and flow rate conservation equations and dynamic equations are used for the movable parts of the system under stress or friction. The magnetic force acting on the electro-injector actuator has been calculated by means of a finite element simulation. The one-dimensional code simulated the propagation in feeding pipes and the control of the electro-injector. The program, in fact, uses the method of the characteristic equations to solve conservation equations, simulating the propagation in a pipe between two chambers. The sensitivity analysis has pointed out that the parameters that are influenced by the propagation in the pipes are: needle lift, injected flow rate, pressure in each chamber, and volume. The perturbations reduce the effective pressure of injection and are influenced by pipe lengths and diameters.

Bifurcation phenomena in steady flows of a viscous fluid II. Experiments

1978 ◽  
Vol 359 (1696) ◽  
pp. 27-43 ◽  
Keyword(s):  
Reynolds Number ◽  
Viscous Fluid ◽  
Experimental Work ◽  
Critical Value ◽  
Taylor Vortex ◽  
Hysteresis Phenomenon ◽  
Steady Flows ◽  
Primary Mode ◽  
Bifurcation Phenomena ◽  
Theoretical Standpoint

The experimental work is concerned with several phenomena studied theoretically in part I (Benjamin 1977). Observations on Taylor-vortex flows between cylinders of comparatively small but variable length are reported, revealing properties unexplained by older theories. The observed flows are classified as follows: (i) the primary mode which is uniquely possible at small values of the Reynolds number R , and which usually develops smoothly with increasing R ; (ii) secondary modes which are possible only above a respective critical value of R , and which are shown to manifest predicted behaviour as this value is approached from above (§4). Two novel and surprising examples of (ii) are reported. A predicted hysteresis phenomenon is confirmed, relating to morphogenesis of the primary mode between two-cell and four-cell forms as the length of the annulus is varied (§5). The experimental results are discussed from a theoretical standpoint in §6.

The non-linear interaction of two disturbances in the thermal convection problem

1962 ◽  
Vol 14 (1) ◽  
pp. 97-114 ◽  
Author(s):  
Lee A. Segel
Keyword(s):  
Rayleigh Number ◽  
Differential Equations ◽  
Equilibrium State ◽  
Thermal Convection ◽  
Critical Value ◽  
Free Boundaries ◽  
Linear Interaction ◽  
Linear Ordinary Differential Equations ◽  
Non Linear ◽  
Convection Problem

In the thermal convection problem with free boundaries, the interaction of two ‘roll’ disturbances is considered. The problem is reduced to a pair of non-linear ordinary differential equations, which should also provide a model for the interaction of two disturbances in more general situations than that for which these equations have been derived. The equations contain several parameters which necessitates a discussion of various possible types of solution. Some representative results are: (1) under certain circumstances, an equilibrium state may be composed of a mixture of a linearly stable disturbance and a linearly unstable disturbance; (2) for the thermal convection problem, when the Rayleigh number is slightly above the minimum critical value, the equilibrium state will contain only one of two linearly unstable disturbances. These and other results are compared with experimental observations.

Preferred pattern of convection in a porous layer with a spatially non-uniform boundary temperature

1993 ◽  
Vol 246 ◽  
pp. 529-543 ◽  
Author(s):  
D. N. Riahi
Keyword(s):  
Porous Layer ◽  
Perturbation Technique ◽  
Three Dimensional ◽  
Critical Value ◽  
Finite Amplitude ◽  
Surprising Result ◽  
Dimensional Solution ◽  
Boundary Temperature ◽  
Steady Solutions ◽  
The One

The problem of finite-amplitude thermal convection in a porous layer between two horizontal walls with different mean temperatures is considered when spatially non-uniform temperature with amplitude L* is prescribed at the lower wall. The nonlinear problem of three-dimensional convection for values of the Rayleigh number close to the classical critical value is solved by using a perturbation technique. Two cases are considered: the wavelength γ(b)n of the nth mode of the modulation is equal to or not equal to the critical wavelength γc for the onset of classical convection. The preferred mode of convection is determined by a stability analysis in which arbitrary infinitesimal disturbances are superimposed on the steady solutions. The most surprising results for the case γ(b)n = γc for all n are that regular or non-regular solutions in the form of multi-modal pattern convection can become preferred in some range of L*, provided the wave vectors of such pattern are contained in the set of wave vectors representing the spatially non-uniform boundary temperature. There can be critical value(s) L*c of L* below which the preferred flow pattern is different from the one for L* > L*c. The most surprising result for the case γ(b)n ≠ γc and γ(b)n ≡ γ(b) for all n is that some three-dimensional solution in the form of multi-modal convection can be preferred, even if the boundary modulation is one-dimensional, provided that the wavelength of the modulation is not too small. Here γ(b) is a constant independent of n.

Sign in / Sign up

Export Citation Format

Share Document