Axisymmetric geometry

The axisymmetrical geometry of the SECM implies that there is no radial flux of the... 

The time-dependent diffusion equations for Red appropriate to the axisymmetrical geometry, shown in Fig. 10, are identical to Eqs. (9) and (10), given earlier. Although phase 2 is assumed to be semi-infinite in the z-direction, the model can readily be modified for the situation where phase 2 has a finite thickness [61]. 

Figure 2 Boundary conditions applied to an axisymmetric geometry.

Step 1 To solve this, enter FEMLAB and choose Chemical Engineering, Axisymmetric 2D/Momentum/Navier-Stokes/Stationary. In axisymmetric geometries, FEMLAB uses the horizontal axis as radius (called r) and the vertical axis as the length axis (called z). The corresponding velocity components are u in the radial direction and v in the axial direction. The sketch shows how the geometry is displayed. 

We assume that we have a thin film. A fixed amount of surfactant is deposited onto the surface of this thin film, either as a 2D strip or as an axisymmetric drop. We assume that this surfactant is insoluble in the bulk fluid of fhe thin film. We denote the total (fixed) mass of surfactant as M, which may be related to fhe surface concentration F (measured in units of mass per unit surface area) in a form that encompasses both the 2D and axisymmetric geometries ... 

To be specific, we denote the three coordinate directions as (qu q2, q2), with q2 being either the axial coordinate z in the 2D case or the azimuthal angle

scale factors (hi, h2, hi), corresponding to qt, defined such that the length of a differential line element can be expressed in terms of increments in the q ... 

Thus, in a manner entirely equivalent to the two-dimensional analysis, we seek rescaled equations in the inner (boundary-layer) region very near to the sphere surface within which the tangential velocity goes from the potential-flow value (3/2)sin0 to 0 at the body surface. The only difference from the previous analysis is in the detailed form of the Navier-Stokes and continuity equations for axisymmetric geometries. When expressed in terms of spherical coordinates, these equations are... 

Figure 3.4 Results for axisymmetric geometry with various initiator mixtures driving a stoichiometric main combustor (C2H4-air, Do = 5.08 cm initial pressure 100 kPa and initial temperature 283 K) 1 — successful 2 — unsuccessful and 3 — computational result. At ( = 0.4, uncertainty is 0.1, and at 1,2 = 1.4, uncertainty is 0.04.

The required initiator strength (equivalence ratio) wcts found to be higher than the axisymmetric case. This is likely due in part to the decrease in confinement at the diffraction plane and the fact that the wall reflection from the 2D geometry does not possess the benefits of wall curvature found on the axisymmetric geometry. The effect of wall curvature has been found to be important for the generation of X -shaped compression waves and increased local heating. 

A simplified calibration test case of the in-situ experiment was proposed and defined below. The case focuses on the THM behaviour of a radial line (with a radial distance r as the coordinate) from the centre of the heater, with the axisymmetric geometry shown in Figure 2. 

A series of numerical simulations of each of the phases of the Isothermal Test have been performed. For these analyses a two-dimensional, axisymmetric geometry has been used as a high degree of radial symmetry was observed in the experimental results. 

Taking the axisymmetric geometry of an unstabilised Z-pinch in which the only magnetic field component is B0, we can form the exact... 

Within the framework of a geometrical approach, the force is calculated from the shape of the meniscus. In simple axisymmetrical geometries, it has two components ... 

Fig. 18.5 Computational domain and initial mesh for simulating the formation and disintegration of conical swirling liquid sheets based on VOF approach, liquid injector highlighted in red (left) 3D geometry (right) 2D axisymmetric geometry...

Figure 18.18 (left) exhibits the calculated gas flow field from an individual gas jet at atomization pressure po = 0.5 MPa (polPa = 5). The diameter at the nozzle exit is 3 mm. The simulation is conducted based on the 2D axisymmetric geometry (see Fig. 18.15). Five cells with shocks can be found after the nozzle exit. Figure 18.18 (right) exhibits the velocity distribution at the centre line of the jet. The experimental data were obtained by laser Doppler anemometry (LDA) [26]. A good agreement is achieved between experimental data and numerical simulation results, for example in the location and number of shock cells, the calculated length of the supersonic core of the jet and the decay rate of the gas velocity in the subsonic region. Only the amplitudes of the velocity fluctuation differ between experiment and simulation the peak in velocity values behind the shock is more intense than those measured in the experiment. The experimental deviation may be caused by the behaviour of the tracer particles used for LDA measurements. These small but still inertial tracer particles cannot follow the steep velocity gradients across a shock exactly. The k-co SST model indicates a better performance than the standard k-e model.

The problem statement is decribed below for the axisymmetric geometry. The numerical method for RHT has been extended to the 3D case in Refs. [34, 35]. 

On the rounded part of the interface the boundary conditions (Eqs. (8.2a) and (8.2b)) were used. In axisymmetric geometry, only interfaces with a horizontal facet can be considered. Consideration of oblique facets requires a three-dimensional simulation. On the other hand, there are situations when several inclined facets of the same type can be formed at the interface. In this case some elements of rotational symmetry are retained and can be used in simulation. Therefore, in order to treat cases similar to those in Ref. [44] a set of oblique facets was replaced by a sole conical facet, which is a conical surface with a cone angle depending on the inclination of the facets and a lateral size depending on the maximum supercooling. Other aspects of the model were the same as described in the previous sections. 

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