Nodal flow

Tanaka, Y., Y. Okada, and N. Hirokawa. 2005. Fgf-Induced Vesicular Release of Sonic Hedgehog and Retinoic Acid in Leftward Nodal Flow Is Critical for Left-Right Determination. Nature 435, no 7039 172-77. 

M. (2002) Conserved function for embryonic nodal cilia. Nature, 418, 37-38 Nonaka, S., Shiratori, H., Saijoh, Y., and Hamada, H. (2002) Determination of left-right patterning of the mouse embryo by artificial nodal flow. Nature, 418, 96-99. 

Hirokawa, N., Tanaka, Y, Okada, Y, and Takeda, S. (2006) Nodal flow and the generation of left-right asymmetry. Cell, 125, 33 5. 

Lagrangian-Eulerian (ALE) method. In the ALE technique the finite element mesh used in the simulation is moved, in each time step, according to a predetermined pattern. In this procedure the element and node numbers and nodal connectivity remain constant but the shape and/or position of the elements change from one time step to the next. Therefore the solution mesh appears to move with a velocity which is different from the flow velocity. Components of the mesh velocity are time derivatives of nodal coordinate displacements expressed in a two-dimensional Cartesian system as... 

Step 1 - assume a Newtonian flow and obtain the nodal pressures. 

Solution of the algebraic equations. For creeping flows, the algebraic equations are hnear and a linear matrix equation is to be solved. Both direct and iterative solvers have been used. For most flows, the nonlinear inertial terms in the momentum equation are important and the algebraic discretized equations are therefore nonlinear. Solution yields the nodal values of the unknowns. 

There are three principal categories of rec tification tests according to Mah Chemical Process Structures and Infoimation Flows, Butter-worths, Boston, 1989, p. 414). These are the global test, the constraint test (nodal test), and the measurement test. There are variations published in the literature, and the reader is referred to the references for discussion of those. 

To carry out a numerical solution, a single strip of quadrilateral elements is placed along the x-axis, and all nodal temperatures are set Initially to zero. The right-hand boundary is then subjected to a step Increase in temperature (T(H,t) - 1.0), and we seek to compute the transient temperature variation T(x,t). The flow code accomplishes this by means of an unconditionally stable time-stepping algorithm derived from "theta" finite differences a solution of ten time steps required 22 seconds on a PC/AT-compatible microcomputer operating at 6 MHz. 

In the discussion above only a single pipe section was considered. For a network, nodal continuity equations in flows and pressures are also required (Nl, W12). If a pipe section is too long to be treated as one cell, it may be divided into as many cells as necessary. Intermediate nodes are introduced between cells and the equations are augmented accordingly. 

In a reactive transport model, the domain of interest is divided into nodal blocks, as shown in Figure 2.11. Fluid enters the domain across one boundary, reacts with the medium, and discharges at another boundary. In many cases, reaction occurs along fronts that migrate through the medium until they either traverse it or assume a steady-state position (Lichtner, 1988). As noted by Lichtner (1988), models of this nature predict that reactions occur in the same sequence in space and time as they do in simple reaction path models. The reactive transport models, however, predict how the positions of reaction fronts migrate through time, provided that reliable input is available about flow rates, the permeability and dispersivity of the medium, and reaction rate constants. 

Fig. 2.11. Configurations of reactive transport models of water-rock interaction in a system open to groundwater flow (a) linear domain in one dimension, (b) radial domain in one dimension, and (c) linear domain in two dimensions. Domains are divided into nodal blocks, within each of which the model solves for the distribution of chemical mass as it changes over time, in response to transport by the flowing groundwater. In each case, unreacted fluid enters the domain and reacted fluid leaves it.

To see why numerical dispersion arises, consider solute passing into a nodal block, across its upstream face. Over a time step, the solute might traverse only a fraction of the block s length. In the numerical solution, however, solute is distributed evenly within the block. At the end of the time step, some of it has in effect flowed across the entire nodal block and is in position to be carried into the next block downstream, in the subsequent time step. In this way, the numerical procedure advances some of the solute relative to the mean groundwater flow, much as hydrodynamic dispersion does. 

There is a close similarity with planar electromagnetic cavities (H.-J. Stockmann, 1999). The basic equations take the same form and, in particular, the Poynting vector is the analog of the quantum mechanical current. It is therefore possible to experimentally observe currents, nodal points and streamlines in microwave billiards (M. Barth et.al., 2002 Y.-H. Kim et.al., 2003). The microwave measurements have confirmed many of the predictions of the random Gaussian wave fields described above. For example wave function statistics, current flow and... 

Instead of nodal lines in closed systems we are interested in the statistics of NPs for open chaotic billiards since they form vortex centers and thereby shape the entire flow pattern (K.-F. Berggren et.al., 1999). Thus we will focus on nodal points and their spatial distributions and try to characterize chaos in terms of such distributions. The question we wish to ask is simply if one can find a distinct difference between the distributions for nominally regular and irregular billiards. The answer to this question is clearly positive as it is seen from fig. 3. As shown qualitatively NPs and saddles are both spaced less regularly in chaotic billiard in comparison to the integrable billiard. The mean density of NPs for a complex RGF (9) equals to k2/A-k (M.V. Berry et.al., 1986). This formula is satisfied with good accuracy in both chaotic and integrable billiards. 

Thermal stress calculations in the five cell stack for the temperature distribution presented above were performed by Vallum (2005) using the solid modeling software ANSYS . The stack is modeled to be consisting of five cells with one air channel and gas channel in each cell. Two dimensional stress modeling was performed at six different cross-sections of the cell. The temperature in each layer obtained from the above model of Burt et al. (2005) is used as the nodal value at a single point in the corresponding layer of the model developed in ANSYS and steady state thermal analysis is done in ANSYS to re-construct a two-dimensional temperature distribution in each of the cross-sections. The reconstructed two dimensional temperature is then used for thermal stress analysis. The boundary conditions applied for calculations presented here are the bottom of the cell is fixed in v-dircction (stack direction), the node on the bottom left is fixed in x-direction (cross flow direction) and y-direction and the top part is left free to... 

The previous section used the constant strain three-noded element to solve Poisson s equation with steady-state as well as transient terms. The same problems, as well as any field problems such as stress-strain and the flow momentum balance, can be formulated using isoparametric elements. With this type of element, the same (as the name suggests) shape functions used to represent the field variables are used to interpolate between the nodal coordinates and to transform from the xy coordinate system to a local element coordinate system. The first step is to discretize the domain presented in Fig. 9.12 using the isoparametric quadrilateral elements as shown in Fig. 9.15. 

Figure 9.27 shows a schematic of the weighting functions around a nodal point. It is clear that the side that lies in the upwind direction of the flow adds more weight that the side that lies in the down-wind direction. 

Based on the flow analysis network, Wang etal., [18] and Osswald [11] developed the finite element/control volume appproach (FEM-CVA) for injection and compression molding, respectively. Similar to FAN, FEM-CVA assigns a fill factor to every nodal point or nodal control volume. The nodal control volumes are constructed by connecting element centroids to element midsides, as shown in Fig. 9.28. 

Melt front nodes - Nodal control volumes containing the free flow front (0 [Pg.493]

The boundary conditions are defined in the same way as with the flow analysis network. The nodes whose control volumes are empty or partially filled are assigned a zero pressure, and the gate nodes are either assigned an injection pressure or an injection volume flow rate. Just as is the case with flow analysis network, a mass balance about each nodal control volume will lead to a linear set of algebraic equations, identical to the set finite element formulation of Poisson s or Laplace s equation. The mass balance (volume balance for incompressible fluids) is given by... 

Once the boundary conditions are applied, the pressure field can be solved using the appropriate matrix solving routines. Note that for mold filling problems, there is a natural boundary condition that satisfies no flow across mold boundaries or shear edges, dp/dn = 0. Once the pressure field has been solved, it is used to perform a mass balance using eqn. (9.144) or (9.145). Once the flowrates across nodal control volume boundaries are known, a simulation program updates the nodal control volume fill factors using... 

The optimum time step in a FEM-CVA simulation is the one that fills exactly one new control volume. Once the fill factors are updated, the simulation proceeds to solve for a new pressure and flow field, which is repeated until all fill factors are 1. While the FEM-CVA scheme does not know exactly, where the flow front lies, one can recover flow front information in post-processing quite accurately. One very common technique is for the simulation program to record the time when a node is half full, / = 0.5. This operation is performed when the nodal fill factors are updated if the node has fk <0.5 and fk+1 >0.5 then the time at which the fill factor was 0.5 is found by interpolating between tk and tk+l. These half-times are then treated as nodal data and the flow front or filling pattern at any time is drawn as a contour of the corresponding half-times, or isochronous curves. 

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