Convection dominated

The first order derivative in Equation (2.80) corresponds to the convection in a field problem and the examples shown in Figure 2.26 illustraTes the ina bility of the standard Galerkin method to produce meaningful results for convection-dominated equations. As described in the previous section to resolve this difficulty, in the solution of hyperbolic (convection-dominated) equations, upwind-ing or Petrov-Galerkin methods are employed. To demonstrate the application of upwinding we consider the case where only the weight function applied to the first-order derivative in the weak variational statement of the problem, represented by Equation (2.82), is modified. 

Brooks, A. N, and Hughes, T. J.R., 1982. Streamline-upwind/Petrov Galerldn formulations for convection dominated hows with particular emphasis on the incompressible Navier -Stokes equations. Cornput. Methods Appl Meek Eng. 32, 199-259. 

Hughes, T. J.R. and Brooks, A.N., 1979, A multidimensional upwind scheme with no cross-wind diffusion. In Hughes, I . J. R. (ed.), Finite Element Methods for Convection Dominated Flows, AMD Vol. 34, ASME, New York. 

The thermal conductivity of polymeric fluids is very low and hence the main heat transport mechanism in polymer processing flows is convection (i.e. corresponds to very high Peclet numbers the Peclet number is defined as pcUUk which represents the ratio of convective to conductive energy transport). As emphasized before, numerical simulation of convection-dominated transport phenomena by the standard Galerkin method in a fixed (i.e. Eulerian) framework gives unstable and oscillatory results and cannot be used. 

If the rate of moisture vaporization is controlled bv the rate of heat transfer to the wet soHd, then for convection dominated heat transfer, h, at the boiling poiat of water, the characteristic time is... 

S. Murakami, S. Kato, S. Nagano, Y. Tanaka. Diffusion characteristics of airborne particles with gravitational settling in a convective-dominant indoor flow field. ASHRAE Transactions. 98(1), 1992, 82-97. 

Figure 5. Temperature contours in convection-dominated flow.

Hughes, T.R.J. Brooks, A. Finite Element Methods in Convection Dominated Flows, American Society of Mechanical Engineers, p. 19-36, 1979. 

Figure 10. Streamlines (top) and relative gas phase composition of A1 species (bottom) In a vertical axlsymmetrlc reactor at five different times during growth of an AlAs/GaAs superlattice. Red corresponds to all A1 species, violet to no A1 species. The corner Insert portrays the variation In solid fraction of A1 across the Interface. Forced convection dominated flow.

Special Numerical Methods for Convection-Dominated Problems... 

Streamline upwind Petrov/Galerkin formulation for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations,... 

The impact of the diffusion coefficient was also analysed here values of isoprene and oxygen were taken [85], It was found that the RTD of one component in a cycling experiment may be narrow, while the other component is not at optimum. By changing the channel diameter, such a sequence may even be reversed. It is advised [85] to operate in the diffusion-dominated branch rather than in the convection-dominated branch. 

Tobias and Hickman (T2), the only investigators to date to study combined free and forced convection in horizontal channel flow, found a remarkably sharp separation between forced- and free-convection dominated mass transfer. In forced convection, the critical Grashof number, based on the diffusion layer thickness, is... 

At steady state, dc/dt = 0, so an expression for c can be obtained immediately. Obviously, close to the surface, transport of c is dominated by diffusion but further away from the surface convection dominates. 

At small values of Pe, the process is diffusion dominated, although flow has some effect. At a Peclet number of 100, the process is convection dominated. 

CDAF - convection dominated accretion flows (Igumenschev et al. 2002, 2003)... 

If we neglect migration, experiments can be performed under conditions of minimal convection, which are thus dominated by diffusion. Since S increases with time t in such a case, nonstationary conditions exist. On the other hand, if convection dominates in the electrolyte bulk, S 7 /( ), and we approach stationary conditions, as far as diffusion is concerned. 

The Peclet numbers are useful for estimating the relative contributions of convection and diffusion to mass and heat transfer. If Pe is large (>10), convection dominates, and a plug-flow model may be appropriate for simple reactor computations. When Pe is small (<<1), diffusion dominates, and the system behaves like a well-stirred reactor. Thus, Pe may be used to estimate whether downstream impurities can diffuse into the deposition zone. 

For convection along a vertical plate, Ra > 109 indicates that turbulent flow is established and heat transfer by convection dominates. For smaller values of the Rayleigh number, Ra < 104, the flow is laminar and conduction dominates. Thus, the Rayleigh number discriminates between conduction and convection [2]. 

If the reaction has little effects on the flow and the convection dominates the mass transfer, then Equations (21)—(24) can be reduced into algebraic expression of subgrid concentration of transferred component, as follows ... 

A.N. Brooks and T.J.R. Hughes. Streamline upwind-Petrov-Garlerkin formulation for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations. Comp. Meth. Appl. Mech. Eng., 32 199, 1982. 

This problem was solved using a one-dimensional RFM with 200 collocation points, evenly distributed along the x-axis using Algorithm 16. Figure 11.6 compares the analytical solution with computed RBFM solutions up to Pe= 100. As can be seen, even for convection dominated cases, the technique renders excellent results. It is important to point out here that for the radial functions method no up-winding or other special techniques were required. 

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