Numerical techniques to address the movement of the flow were already developed in the FreeBoundary Program (FBP) [2-4]. Numerical simulations can track the advancement of the resin front promoted by both hydrodynamic pressure gradient and capillary action. Base analysis is solved in commercial code ANSYS. However, capillary action implementation brings numerical difficulties, when continuous Galerkin method is used. This can be over- come by post-processing the free-boundary normal velocities, resulting in superior conver- gence results. Post-processing a finite element solution is a well-known technique, which consists in a recalculation of the originally obtained quantities such that the rate of conver- gence increases without the need for expensive remeshing techniques [6-9]. Post-processing is especially effective in problems where better accuracy is required for derivatives of nodal variables in regions where Dirichlet essential boundary condition is imposed strongly . Consequently, such an approach can be exceptionally good in modeling of resin infiltration under quasi steady-state assumption, because only free-front normal velocities are necessary to advance the resin front to the next position.
Post-processing a finite element solution is a well-known technique, which consists in a recalculation of the originally obtained quantities such that the rate of convergence increases without the need for expensive remeshing techniques. Post- processing is especially effective in problems where better accuracy is required for derivatives of nodal variables in regions where Dirichlet essential boundary condition is imposed strongly. For Darcy flow analogy with thermal analysis can be exploited and technique already published in [5-9] can be implemented. Nevertheless in our case the problem must be posed differently, as explained in next section, with the help of several numerical examples. For Stokes flow new technique, presented in its preliminary form in [10-11], is suggested. Post- processing implementation ensure freeboundary velocities of sufficient precision even from coarse meshes, which can significantly reduce CPU time. Time loss required for recalculation is completely equilibrated by the fact that this way the new free front shape is more exact, smoother and consequently larger time steps will be possible to implement. In summary, post-processing implementation ensure faster calculation without the danger of freeboundary oscillation.
verified directly on ANSYS fluid element FLUID 141, where pressure and velocity components are nodal variables. Two test problems for unit viscosity and mass free fluid are specified in Fig. 6, results of original and recalculated normal velocities are shown in Figs. 7 and 8, respectively. No units are stated in the test problems, because only relative comparison is important. Moreover pressure in these test problems does not correspond to the capillary pressure, because the aim is only to test the efficiency of such methodology. Also here meshes of quad elements were used, now as 5x5, 10x10 and 50x50. The 50x50 mesh results can be assumed as the “exact” solution. In the legend of Figs. 7 and 8 original values of normal velocities are designated as “vy 5“, “vy 10“ and “vy 50“ on 5x5, 10x10 and 50x50 quad meshes, respectively, and the recalculated values are stated as “vy-calc 5“ and “vy-calc 10“.
We have formulated the governing equations for the freeboundaryflows in intra- as well as inter-tow spaces and developed numerical techniques to address the movement of the flow at the mesolevel scale, which we call the FreeBoundary Program (FBP). Numerical simulations can track the advancement of the resin front promoted by both hydrodynamic pressure gradient and capillary action [2, 7, 12-14]. Quasi steady state assumption can be exploited in the full flow domain and explicit time integration is adopted along the time scale. FBP is thus concerned with the moving flow front, which requires results at time t k ; approximation of the front at t k
We have formulated the governing equations for freeboundaryflows in intra- as well as inter- tow spaces and developed numerical techniques to address the movement of the flow at the mesolevel scale which we call the FreeBoundary Program (FBP). Numerical simulations can track the advancement of the resin front promoted by both hydrodynamic pressure gradient and capillary action [2-5]. In such simulations it is extremely important to account correctly for the surface tension effects, which can be modeled as capillary pressure applied at the flow front. Unfortunately essential boundary conditions of this kind make the problem ill-posed, in terms of the weak classical as well as stabilized formulation. As a consequence there is an error in mass conservation accumulated especially along the free front. This can affect significantly normal velocities at the free front and distort the next front shape. Due to the explicit integration along the time scale, such errors are irreversible. Several stabilization techniques were implemented in FBP to eliminate this effect [3-5]. In this article we will present more appropriate techniques for stabilization, based on weak formulation of the problem. The methodology implemented in Darcy’s region is well-known, although rarely used in real simulations. It is presented e.g. in . The recalculated outlet velocities have superior convergence properties . In Stokes region the correction of the outlet velocities we are presenting have not yet been published to our knowledge. Both methodologies are implemented in FBP.
It is possible to characterize the turbulence by its two main measures: intensity and scale, usually related to a velocity along an average stream line. The influence of turbulence intensity on transition is quite well known. The formulas describing the relation between the intensity and the onset Reynolds number are given for example by Mayle (1991), Hall and Gibbings (1972) or Abu-Ghannam and Shaw (1980). But there are still very few investigations relating to the influence of the turbulence scale on laminar–turbulent transition. Mayle (1991), in his review, suggested that the transition appears earlier when the mesh of the grid is smaller (what implies a smaller length scale). Also Jonas et al. (2000) suggested that the inception and the transition length depend on the turbulence scale. Their experimental results indicate that the onset of bypass transition moves downstream with decreasing length scale of turbulent disturbances at a fixed intensity of turbulent fluctuations in the leading edge plane – the laminar part of the boundary layer becomes longer. The transition region becomes shorter. Nevertheless, the transition process terminates earlier in flow with larger turbulence length scale than in flow with a smaller value of it. The turbulence intensity at the leading edge of the plate was maintained constant (Tu = 3%), whilst the values of the dissipation length scale were changing: Lu 2 . 2 ; 33 . 3 mm. The outcome of the investigation was a following correlation:
We study the freeboundary problem for the“hard phase” material introduced by Christodoulou in , both for rods in (1 + 1)-dimensional Minkowski spacetime and for spherically symmetric balls in (3+1)-dimen- sional Minkowski spacetime. Unlike Christodoulou, we do not consider a “soft phase”, and so we regard this material as an elastic medium, capable of both compression and stretching. We prove that shocks, defined as hypersurfaces where the material’s density, pressure and velocity are discontinuous, must be null hypersurfaces. We solve the equations of motion of the rods explicitly, and we prove existence of solutions to the equations of motion of the spherically symmetric balls for an arbitrarily long (but finite) time, given initial conditions sufficiently close to those for the relaxed ball at rest. In both cases we find that the solutions contain shocks if the pressure or its time derivative do not vanish at the freeboundary initially. These shocks interact with the freeboundary, causing it to lose regularity.
For every flight presented, we determined the height of the planetary boundary layer (PBL) based on in-situ CO measurements with a time resolution of 1 s. As an example, Fig. 8 shows the vertical profile of CO, measured on 25 October 2008. Below 500 m height a.g.l. the CO mixing ratio ranges from 200 to 250 ppb, this height regime is inter- preted as the well mixed PBL. Between 500 and 750 m a very sharp decrease from 200
In this paper the main result is the asymptotic behavior of the freeboundary. We remark that the upper bound (4.47) should be very useful for real applications, where the function f is a priori unknown and a estimate of s ∞ is needed. From the physical point of view we emphasize that the bound of the freeboundary does not depend on the function f . That means that this behavior of the freeboundary holds for all kind of homogeneous polymers with constant diffusivity. For the case of two dimensional space variable we expect to have bounds for the freeboundary that do not depend on f. This will be the subject of future work.
Meshfree methods were introduced to eliminate part of those difficulties such as distorted ele- ments, need for re-meshing and the other limitations which arise because of element connectivity. A large variety of meshfree methods have been developed up to now for solving different engineering problems, such as smooth particle hydrodynamics (SPH) by Lucy (1997), diffuse elements, Nayroles et al. (1992), Element-Free Galerkin method (EFG), Belytschko et al. (1994), reproducing kernel particle method (RKPM), Liu et al. (1995), and hp-clouds methods, Duarte and Oden (1995).
where N corresponds to . Here the suffix i corresponds to and j corresponds to . Also = j 1 j and = i 1 . Knowing the values of i , u at a time we can calculate the values at a time as follows . We substitute = 1, 2,..., i N , in equation (10) which constitute a tri-diagonal system of equations, 1 the system can be solved by Thomas algorithm as discussed in (Carnahan et al., 1969). Thus is known for all values of at time . Then knowing the values of and applying the same procedure with the boundary conditions, we calculate, u from equation (9). This procedure is continued to obtain the solution till desired time . The Crank-Nicolson scheme has a truncation error of 2 2
Four numerical simulations were done for the boundary- layer configuration. The boundary-layer configuration is sim- ilar to the open-channel configuration but not exactly the same. Like the open-channel configuration, a slip boundary condition is imposed at the top wall for the fluid phase. In Fig. 7, bulk streamwise velocity, steady-state sediment con- centration profile and normalized turbulence intensity have been plotted. The four simulations for the open-channel-like configuration have the same particle fall velocity ( ˜ V ) but increasing shear Richardson number. Similar to the chan- nel flow configuration, bulk streamwise velocity was found to increase with increase in shear Richardson number. Tur- bulence intensity was found to decrease with increase in shear Richardson number, and unlike the channel flow con- figuration where turbulence intensity decreases in the lower half of the channel and increases in the upper half; turbu- lence intensity was found to decrease throughout the chan- nel. Though, the extent of damping in the upper half was found to be slightly more than the extent of damping in the lower half of the boundary layer. The steady-state sed- iment concentration profiles in Fig. 7 were used to calcu- late vertical sediment diffusivity profiles (Fig. 8) for the four cases simulated. Reflecting the trend shown by turbu- lence intensity, vertical sediment diffusivity was found to de- crease with increase in level of self-stratification. The sedi- ment diffusivity profiles were quantified using Eq. (10) and the parameters K zµ , K zσ and K zγ have been listed in Ta-
Numerical simulation of nonlinear wave interaction with a fixed structure at the two dimensional NURBS NWT is conducted in this paper. MEL method combined with high order boundary ele- ment method is applied to model the free surface distortion. High order boundary integral equation based on Green’s second identity and NURBS curve is employed to solve Laplace equation in the Eulerian frame. To obtain free surface elevation and re-gridding, NURBS interpolation is used. To update fully nonlinear free surface boundary condition in the Lagrangian manner, material node approach and fourth order Rung-Kutta time integration scheme are employed. Double node tech- nique is applied to overcome the difficulties arising from the singularity at the corner points.
ern Chile reasonably well, and performs better than some previous studies (Wang et al., 2009). This may come from the inclusion of ultrafine sea salt emissions, and the boundary layer nucleation mechanism in the MMF. However, the model slightly un- derestimates the Aitken mode number concentration over Scotland in the lower free troposphere, which may be partly explained by the fact that carbonaceous aerosols
the Hoppel minimum. These plots also indicate a smaller fraction of aerosol that did not act as CCN in the MBL when CO is higher. This is consistent with more FT aerosol be- ing larger and more likely to be CCN compared to the low CO cases (Fig. 1). However, it is important to note that the BL distributions represent the net effect of entrainment that took place upwind of measured profiles while the FT data are yet to be entrained and will only influence the MBL over the next several days. We have also not yet accounted for sea salt present in the MBL (Hudson et al., 2011) or the observations of vertical mixing near the end of PASE when winds and CO values were generally higher and when boundary layer rolls were more prevalent. We also note that size distributions in Fig. 10 can involve sampling of cloud outflow mixed with preexisting aged FT aerosol, probably derived from similar processes in the past. Hence, both groupings include con- tributions from preexisting FT aerosol so they are less well defined than distributions for individual cases (Fig. 1).
of HAT in El Ni ˜no and La Ni ˜na years are found over all the circumpolar intercontinental routes for transpacific, transatlantic and transeurasian transport of aerosol in the NH scale (Fig. 11a,b). It is also revealed that a strengthening and southward shift of west- erly jet not only over the North Pacific but also in the entire mid-latitude westerly ozone is evident for the HAT during El Ni ˜no winter and spring. In the boundary layer, the more
it into (2.4), implying an evolution equation for the free surface, h(x, t). Therefore, the solution to our problem goes through a Dirichlet-to-Neumann operator. We will, however, avoid explicitly introducing it here, for the sake of simplicity and as we would not really gain much in doing so. Let us also point out here that the main diﬀerence between the equations governing the classical water-wave problem and those that govern the porous medium problem under consideration can be seen in (2.2), which in the last case is much simpler than in the former case, where it involves time- derivatives and nonlinear terms.
times increase, and Brent with a 4.3 times increase. Three of these four fields have GORs near or in excess of 20,000 scf/bbl (see SI worksheet for data inputs), implying large energy require- ments for gas processing in the default configuration. These variations should be considered part of the uncertainty associated with modeling a field where the specific processing configu- ration is not known by the modeler. Because required product specifications and the quality of the underlying resource are given quantities, the choice of processing configuration is not one set completely by free operator choice (e.g., operators cannot generally choose to avoid desul- furizing sour gas to avoid the energy cost of acid gas removal, as the resulting gas would not meet quality specifications).
The MABL is the atmospheric surface layer above the ocean in which trace gas emis- sions are mixed vertically by convection and turbulence on short time scales of about an hour (Stull, 1988; Seibert et al., 2000). The upper boundary of the MABL is ei- ther limited by a stable layer e.g. a temperature inversion or by a significant reduc- tion in air moisture. Determination of the MABL height can be achieved by theoretical