Outflow boundary condition

To avoid imposition of unrealistic exit boundary conditions in flow models Taylor et al. (1985) developed a method called traction boundary conditions . In this method starting from an initial guess, outflow condition is updated in an iterative procedure which ensures its consistency with the flow regime immediately upstream. This method is successfully applied to solve a number of turbulent flow problems. 

Renardy, M., 1997. Imposing no boundary condition at outflow why does it work Int. J. Numer. Methods Fluids TA, 413-417. 

One nice feature of the finite element method is the use of natural boundary conditions. It may be possible to solve the problem on a domain that is shorter than needed to reach some limiting condition (such as at an outflow boundary). The externally applied flux is still applied at the shorter domain, and the solution inside the truncated domain is still valid. Examples are given in Chang, M. W., and B. A. Finlayson, Inf. J. Num. Methods Eng. 15, 935-942 (1980), and Finla-son, B. A. (1992). The effect of this is to allow solutions in domains that are smaller, thus saving computation time and permitting the solution in semi-infinite domains. 

A convenient concept for introducing the surface boundary condition into the mathematical formulation of migration theory is that of what may be called a diffusional offset length d. Suppose that the external and surface conditions are describable by a set of parameters X, which we do not need to specify in detail we also allow the surface conditions to depend on the internal hydrogen concentration just beneath the surface. If the hydrogen complexes that are continually forming in the crystal are sufficiently immobile, the balance between inflow and outflow across the surface will depend only on X and on the concentration no(0) of H0 just beneath the surface. (If mobile H+ or H are present, the statement just... 

Simulations are then performed for gas bubbles emerging from a single nozzle with 0.4 cm I.D. at an average nozzle velocity of lOcm/s. The experimental measurements of inlet gas injection velocity in the nozzle using an FMA3306 gas flow meter reveals an inlet velocity fluctuation of 3-15% of the mean inlet velocity. A fluctuation of 10% is imposed on the gas velocity for the nozzle to represent the fluctuating nature of the inlet gas velocities. The initial velocity of the liquid is set as zero. An inflow condition and an outflow condition are assumed for the bottom wall and the top walls, respectively, with the free-slip boundary condition for the side walls. 

Subsequently, simulations are performed for the air Paratherm solid fluidized bed system with solid particles of 0.08 cm in diameter and 0.896 g/cm3 in density. The solid particle density is very close to the liquid density (0.868 g/ cm3). The boundary condition for the gas phase is inflow and outflow for the bottom and the top walls, respectively. Particles are initially distributed in the liquid medium in which no flows for the liquid and particles are allowed through the bottom and top walls. Free slip boundary conditions are imposed on the four side walls. Specific simulation conditions for the particles are given as follows Case (b) 2,000 particles randomly placed in a 4 x 4 x 8 cm3 column Case (c) 8,000 particles randomly placed in a 4 x 4 x 8 cm3 column and Case (d) 8,000 particles randomly placed in the lower half of the 4x4x8 cm3 column. The solids volume fractions are 0.42, 1.68, and 3.35%, respectively for Cases (b), (c), and (d). 

In the early days, see, e.g., Bakker and Van den Akker (1994a), a black box representing the impeller swept volume was often used in RANS simulations, with boundary conditions in the outflow of the impeller which were derived from experimental data. The idea behind this approach was that such nearimpeller data are hardly affected by the rest of the vessel and therefore could be used throughout. Generally, this is not the case of course. Furthermore, this approach necessitates the availability of accurate experimental data, not only... 

The boundary conditions are otherwise zero flux at the walls and outflow conditions at the outlet(s). 

During the MC simulation, boundary conditions must be applied at the edges of the flow domain. The four most common types are outflow, inflow, symmetry, and a zero-flux wall. At an outflow boundary, the mean velocity vector will point out of the flow domain. Thus, there will be a net motion of particles in adjacent grid cells across the outflow boundary. In the MC simulation, these particles are simply eliminated. By keeping track of the weights... 

The response of a single complete mix reactor to a pulse or front input is essential to our analysis. A pulse would be a Dirac delta in concentration, whereas a front would be a sharp step at a given time to a different concentration. Let us first develop the outflow equation for a front input, illustrated in Figure 6.3, with boundary conditions ... 

Originally this problem is formulated in a semi-infinite channel. In our numerical computations we have considered a finite one of length 2Lr. At the outflow we have imposed a homogeneous Neumann boundary condition... 

The boundary conditions for the stream-function-vorticity system requires specifying the stream function on all the boundaries. This is usually straightforward for known inflow and outflow conditions and solid walls. The vorticity boundary conditions comes from evaluating Eq. 3.281 on the boundary. Along the boundary, which usually corresponds with one of the coordinate directions, one of the terms in Eq. 3.281 (i.e., the one in which the derivatives align with the boundary) can be evaluated explicitly since the stream function is already specified. Thus the boundary conditions becomes a relationship between the boundary vorticity and a boundary-normal second derivative of stream function. For example, consider the natural convection in a long horizontal tube. Here, since there is no inflow or outflow, the stream function is simply zero all around the tube wall. Thus the vorticity boundary conditions are... 

Typically, there are two types of boundaries in reacting flows. The first is a solid surface at which a reaction may be occurring, where the flow velocity is usually set to zero (the no-slip condition) and where either a temperature or a heat flux is specified or a balance between heat generated and lost is made. The second type of boundary is an inflow or outflow boundary. Generally, either the species concentration is specified or the Dankwerts boundary condition is used wherein a flux balance is made across the inflow boundary (64). The gas temperature and gas velocity profile are usually specified at an inflow boundary. At outflow boundaries, choices often become more difficult. If the outflow boundary is far away from the reaction zone, the species concentration gradient and temperature gradient in the direction of flow are often assumed to be zero. In addition, the outflow boundary condition on the momentum balance is usually that normal or shear stresses are also zero (64). 

The Peclet number is defined as the product of the Reynolds number with the Prandtl number. For a large Peclet number near the outlet, there is no upstream influence because the flow is dominated by the downstream convection. In this case, no boundary condition information is needed for the outflow boundary [Patankar, 1980]. 

These forms relate the dependence on the system characteristics. Equation (8.13) describes the concentration c(z, t) of a solute in a tree-like structure that corresponds to the arterial tree of a mammal. Considering also the corresponding venular tree situated next to the arterial tree and appropriate inflow and outflow boundary conditions, we are able to derive an expression for the spatiotemporal distribution of a tracer inside a tree-like transport network. We also make the assumption that the arterial and venular trees are symmetric, that is, have the same volume V then, the total length is L = V/Ag The initial condition is c(z, 0) = 0 and the boundary conditions are ... 

Inflow or outflow can be specified using appropriate boundary conditions, although some characteristic problems are more easily implemented with one than the other. In any case, it is important to insure that the conditions specified are self-consistent. For example, inflow can be specified along the top boundary, by specifying the inflow velocity at V, as in figure (3). This in turn specifies a relative change in vorticity from to o ... 

The above example of specifying V, and consequently a change from to requires a concomitant reversal in some other region of the boundary and thus outflow. As can be seen, such boundary conditions must be implemented with some care if the numerical scheme is to be self-consistent. 

The numerical jet model [9-11] is based on the numerical solution of the time-dependent, compressible flow conservation equations for total mass, energy, momentum, and chemical species number densities, with appropriate in-flow/outflow open-boundary conditions and an ideal gas equation of state. In the reactive simulations, multispecies temperature-dependent diffusion and thermal conduction processes [11, 12] are calculated explicitly using central difference approximations and coupled to chemical kinetics and convection using timestep-splitting techniques [13], Global models for hydrogen [14] and propane chemistry [15] have been used in the 3D, time-dependent reactive jet simulations. Extensive comparisons with laboratory experiments have been reported for non-reactive jets [9, 16] validation of the reactive/diffusive models is discussed in [14]. 

At the inflow and at the top of the computational domain one calculates the flow variables, as induced by the freestream vortex via Biot-Savart interaction rule. At the outflow, fully developed condition is applied for the wall-normal component of the velocity ( = 0) and using the same in SFE, one can obtain the vorticity boundary condition at the outflow from Equation (3.4.2). At the top frame of Fig. 3.8, one sees incipient unsteady separation on the wall. In subsequent frames, one notices secondary and tertiary events induced by the primary instability. In these computed cases, one does not notice TS waves and the vortices formed on the wall are essentially due to unsteady separation that is initiated by the freestream convecting vortex. These ensemble of events have been noted as the vortex-induced instability or bypass transition in Sengupta et al. (2001, 2003), Sengupta Dey (2004) and in Sengupta Dipankar (2005). 

For Maxwell models in the supercritical case (i.e., U > iJi]l(pX)), the previous choice of boundary conditions leads to an ill-posed problem (as does the Dirichlet boundary condition for a hyperbolic equation), as shown in [17]. In addition to the normal velocities at both boundaries (inflow and outflow) and to the previous inflow conditions on the stresses, one can prescribe the vorticitj and its normal derivative in two space dimensions, or the second and third components of the vorticity and their normal derivatives in three... 

A discussion of the traction boundary conditions—where the totaJ normal stress is prescribed on the inflow and outflow boundaries—for Jeffreys-type fluids is given in [31], and for Maxwell-type fluids in [32]. 

R. Talhouk, Unsteady flows of viscoelastic fluids with inflow and outflow boundary conditions, Appl. Math. Letters, (1996) to appear. 

All parameters have the same definitions as used previously. The initial condition is that drug concentration in the tissue is everywhere zero. The boundary conditions are, first/ that the drug concentration remains zero at all times far from the cannula tip and/ second/ that the mass outflow from the cannula be equal to the diffusive flux through the tissue at the cannula tip/ that iS/ that... 

Click on a boundary number (1-4) (Note The corresponding boundary is highlighted in red.). Set the boundary condition for each boundary segment. 1, slip/symmetry 2, inflow/outflow,... 

Jacobs, G. B., D. A. Kopriva, and F. Mashayek. 2002. Outflow boundary conditions for the multidomain staggered-grid spectral method. AIAA Paper No. 2002-0903. 

Inlet swirl inflow conditions are discussed below. The outflow boundary conditions at the combustor outlet involve advection of all flow and species variables with Uc, where the instantaneous mean streamwise outlet boundary velocity Uc is periodically renormalized to ensure that the time-averaged mass flux coincides with that at the inlet these convective boundary conditions are enforced in conjuction with soft relaxation of the outflow pressure to its ambient value. Two types of outlets were considered (Fig. 11.1). Viscous wall regions in the combustor cannot be practically resolved for the moderately-high Reynolds... 

In ceramic extrusion we typically have to deal with inflow- and outflow boundaries, (moving) walls and free surface boundaries for which the appropriate mathematical formulation will be given in the subsequent paragraphs. Since we use pressure, temperature and velocity as our independent set of variables, the boundary conditions must be expressed in terms of p, T, V. They can take two different forms The dependent variables are specified along the boundary (Dirichlet boundary condition) or the directional derivatives of the dependent variables are prescribed Neumann boundary condition). 

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