Time-Dependent Ferrofluid Dynamics in Symmetry Breaking Transverse

Sebastian Altmeyer

Max-Planck-Institute for Dynamics and Self-Organization, G?ttingen, Germany.

DOI: 10.4236/ojfd.2013.32015                

Abstract

We investigate the Taylor-Couette flow of a rotating ferrofluid under the influence of symmetry breaking transverse magnetic field in counter-rotating small-aspect-ratio setup. We find only changing the magnetic field strength can drive the dynamics from time-periodic limit-cycle solution to time-independent steady fixed-point solution and vice versa. Thereby both solutions exist in symmetry related offering mode-two symmetry with left-or right-winding characteristics due to finite transverse magnetic field. Furthermore the time-periodic limit-cycle solutions offer alternately stroboscoping both helical left-and right-winding contributions of mode-two symmetry. The Navier-Stokes equations are solved with a second order time splitting method combined with spatial discretization of hybrid finite difference and Galerkin method.

Keywords

Taylor-Couette Flow; Ferrofluids; Reynolds Number; Symmetry Breaking; Rotating System and Boundary Layer

1. Introduction

A diffusion flame is a very common practical system, such as a match flame or a candle flame, and is fundamental to many more complex systems. However, in a diffusion flame there is a relatively long time scale, allowing sufficient recombination reactions to take place such that it involves complex interaction of flow, transport and chemistry. There is more chemical and physical interaction in diffusion flames than in a premixed flame. The first systematic analysis of a confined, jet diffusion flame dates back to 1928 by Burke and Schumann (published in 1948) [1]. Due to environmental concerns, the amount of NO_x emission in combustion processed and its formation mechanism have always been of interest. Being of particular interest is source of nitric oxide, which is a major pollutant to the atmosphere. It has been postulated that there are two major mechanisms Zeldovich thermal and Fennimore prompt NO in the production of total NO. Barlow and Carter (1993) [2] reported a rather complete simultaneous measurement on temperature and species concentrations (NO, OH, and major species) in hydrogen flames. However, the amount of naturally generated NO in a gas phase hydrocarbon flame is very small, and most measurements were done with a certain amount of NO addition.

Dong et al. [3] studied the characteristics of an impinging inverse diffusion flame jet the length of the jet impingement region, the characteristic of the wall gauge static pressure and the heat flux in the impingement region they provide a correlation between heat transfer performance and the hydrodynamic behavior, which facilitates the optimization of the jet impingement system design and operation. The aim is to understand the characterization of the extinction limits of fuel-air mixtures from low extinction strain rate methane-air flames to high extinction strain rate ethylene-air flame. Sarnacki et al. [4] reported an experimental and computational work, and they found that the variation of local extinction strain rate with changes in separation distance was within uncertainty of the experimental data. Zhen et al. [5] investigated the thermal and heat transfer behaviors of multifuel jet inverse diffusion flame with induce swirl and non-swirling under identical air/fuel rates. Flame appearances, temperature fields and wall static pressures were examined, and they found that the main reaction zone in the swirling flame is closer to the burner exit and the flame length is much shorter than the non-swirling flame and the wall static pressure and radial heat flux both are influenced by the swirl effect. A large eddy simulation technique was used to study the fuel variability on the dynamics of hydrogen and syngas impinging flames by Mira Martinez et al. [6] including a pure H₂ and (20% CO, 80% H₂), (40% CO, 60% H₂), (20% CO, 20% CO₂, 60% H₂) the results of their study show that the flames develop vertical structures in the primary jet associated with the buoyancy and shear layer instability, and the wall jet progresses parallel to the impinging plate forming large scale vortex rings at different locations and strengths as a consequence of the fuel compositions.

Choe McDaid et al. [7] studied the effect of ignition location on the propagation of premixed and diffusion flames of hydrogen and mixtures of hydrogen and carbon dioxide towards. They found that the fuel velocity and Reynolds had a large effect on the observed flame velocities, and the type of the fuel affected the velocities and accelerations of the flame front. Experimental and numerical study had been conducted by Subhash et al. [8] to investigate the occurrence of off stagnation peak for laminar/air flame impinging on a flat surface of this off stagnation peak using a commercial CFD code Fluent. They found that this off stagnation in heat flux is primarily due to the peak in the axial velocity close to the impingement surface. Jaramillo et al. [9] applied the DNS and RANS techniques to study the fluid flow and heat transfer in plane impinging jets. The DNS results have been used as reference solution to assess the performance of several Reynolds averaged Navier Stokes (RANS) models. However all the models predict correctly the local Nusselt number at the stagnation region to investigate the complex flow field of an impinging jet Naseem et al. [10] used an LES simulation with a dynamic smagorinsky model. The large eddy simulation helps to understand the reason of occurrence of second peak in the radial distribution of Nusselt number at the target wall. They found that the LES simulation of a complex flow impinging jet is highly sensitive to the quality of the grid.

Zhen et al. [11] studied the emission of CO and NO_x from swirling and non-swirling of an impinging inverse diffusion flames and they found that the parameters of air jet Reynolds numbers, overall equivalence ration and Nozzle to plate distance have significant influence on the overall pollutants emission. To reduce the NO_x production a new down fired combustion technology based on multiple injection and multiple staging was developed by Min Kuang et al. [12]. An experimental study was performed by Zhen et al. [13] to investigate the effects of the nozzle length on the air pollutant emission and noise radiation. They found that the noise radiation from the inner reaction cone of the flame is stronger than that from the lower and upper parts of the flame for the stoichiometric air/fuel ration. Zhen et al. [14] performed an experimental work to compare between the emission and impingement heat transfer behaviors liquefied petroleum gas added of hydrogen and methane air flames

(LPG-H₂-air) and (CH₄-H₂-air) comparison shows a more significant change in the laminar burning speed, temperature and CO/NO_x emissions in the CH₄ flames. Gurpreet et al. [15] investigated the heat transfer charactiristics of natural gas/air swirling flame impinging on a flat surface the dip in heat flux at and around stagnation point was observed in almost all cases which could be the main cause of non uniformity even in case of heating with swirling impinging flames.

Nadjib et al. [16] investigated the influence of the nitrogen dilution on the extinction of methane impinging diffusion flame, and they found that when the dilution rate increased the extinction of the diffusion flame increased.

In the current study, we investigated an impinging diffusion flame with three fuels, Methane, Propane and Butane with fixed fuel jet velocity. We also reported on the temperature response to the increase of the CH atom mole fraction, on the other hand we studied the relation between NO fraction production and heating high temperature for a non-premixed turbulent hydrocarbons jet flame situation.

2. Mathematical Model

2.1. Turbulent Governing Equations

In the present study, Fluent [20] (commercial CFD software) was used to model the flow field and heat transfer for diffusion turbulent methane/air flame impinging vertically on a flat surface with a reduced reaction mechanism (8 species are considered).

The general form of transport equations for two dimensional stationary turbulent reactive flows can be written as:

(1)

(2)

where denotes 1, and diffusion coefficient

is 0, and respectively. is a source term. Fluent uses a control-volume-based technique to convert the governing equations to algebraic equations that can be solved numerically.

In order to resolve the turbulent flow problem we are used K-Epsilon turbulent RNG based model.

(3)

(4)

where and

The model constants appearing in the above equations are

and.

The effects of the mean strain rate and mean rotation on turbulent diffusion have been affected by using the renormalized RNG k- model Yakhot et al., 1992 [21], which employs equations of the same form as the standard k- model. The RNG k- model assumes different model coefficients evaluated by the renormalization group theory which vary with the ration of the turbulent to the mean strain, n, as described below:

(5)

with

where

(6)

Though the modification of the above constants of the model, it is intended to simulate and control the modelling of the energy dissipation. The RNG k- model is a modification version of the standard k- turbulent model. It adopts a non-equilibrium strain parameter, Where S is the strain rate modulus and the ratio is the turbulence time scale.

2.2. Modeling Non-Premixed Combustion

In non-premixed combustion, fuel and oxidizer enter the reaction zone in distinct streams. This is in contrast to premixed systems, in which reactants are mixed at the molecular level before burning. Examples of non-premixed combustion include methane combustion, pulverized coal furnaces, and diesel (compression) internalcombustion engines.

In order to resolve the turbulent chemistry interaction we are focused to use Pre PDF model based on the resolution of mean transport species equation and its variance

[20].

(7)

(8)

The species fractions considered in this investigation are

In order to model nitric oxide formation in a flame, the chemical reactions involving nitrogen compounds must be taken into account. The reactions

are the principle mechanisms in forming Zeldovich “thermal” NO; the reactions are the major paths in forming Fenimore “prompt” NO. In the numerical work, both Zeldovich thermal and Fenimore prompt formation of NO were included.

2.3. Computational Domaine

Using a 2D model, the impingement surface is parallel to the fuel jet; and the jet was spreading vertically on the impinging plate. The diameter of the fuel jet is 10 mm Figure 1 a total of seven transport equations (continuity, axial, and radial momentums, turbulence kinetic energy and its dissipation rate, energy, and radiative intensity), are solved using the commercial CFD package FLUENT [20]. A second-order discretization scheme was used to solve all governing equations. Solution convergence was determined by two criteria. First is ensuring that the residuals of the solved equations drop below specified thresholds set at 10⁻³ for all variables, while a residual of 10⁻⁶ was used for the energy equation. The second convergence criterion is ensuring that the value of a sensitive property (e.g., concentration of a radical species) at a critical spatial location has stabilized and is no longer changing with iterations.

Figure 1. Schematic of mesh domain and boundary conditions for jet diffusion flame.

3. Numerical Results

In this simulation the temperature and concentration of major species C₄H₁₀, C₃H₈, CH₄, H₂, H₂O, CO₂, N₂, and O₂ and minor species NO, CO, and OH was performed using Fluent software code. CH₄, C₃H₈, C₄H₁₀, was used as the fuel in the impinging jet.

Velocity magnitude, flame temperature and the structure of the turbulent flame region were compared with different fuel jet flames.

Figures 2-6 show contours plots for temperature and mass fractions of OH and CO₂ and OH and NO and production rate plotted versus mixture fraction for flames C₄H₁₀, C₃H₈, and CH₄ at the same axial location.

The effect of N₂ dilution level, in the fuel stream, on the flame temperature is quite substantial with a drop from 1700 K in the case propane flame, to 1400 K in case of butane flame. The temperature at the centerline is 420 K and is the same for all cases. The mean temperature in the jet vicinity of the flow is 1200 K and is also consistent between all flames Figure 2 this result is compared with the experimental data in references [16-19] for the same authors. When we look in Figure 3 we observe that the mass fraction of CO₂ is maximal for the butane jet flame. The same results are obtained in Figure 4.

Figures 7 and 9 show how the calculated NO and CO₂ concentrations vary with the three jet fuels for the same simulation conditions.

Figures 7-10 show radial profiles of temperature and mass fractions of OH and CO and NO and turbulent intensity for turbulent flames C₄H₁₀, C₃H₈, and CH₄ at an axial position of 100 mm above the jet exit.

Figure 7 shows radial profiles of mean NO mole fractions for the same flames and positions as those in Figure 10 the NO distribution for flame C₄H₁₀ is different

Conflicts of Interest

The authors declare no conflicts of interest.

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