Mean pressure field

A governing equation for the mean pressure field appearing in (2.93) can be found by Reynolds averaging (1.29). This leads to a Poisson equation of the form... 

The mean pressure field (p) appears as a closed term in the conditional acceleration (A, V, ip). Nevertheless, it must be computed from a Poisson equation found by taking the divergence of the mean velocity transport equation ... 

Since the mean velocity and Reynolds-stress fields are known given the joint velocity PDF /u(V x, t), the right-hand side of this expression is closed. Thus, in theory, a standard Poisson solver could be employed to find (p)(x, t). However, in practice, (U)(x, t) and (u,Uj)(x, t) must be estimated from a finite-sample Lagrangian particle simulation (Pope 2000), and therefore are subject to considerable statistical noise. The spatial derivatives on the right-hand side of (6.61) are consequently even noisier, and therefore are of no practical use when solving for the mean pressure field. The development of numerical methods to overcome this difficulty has been one of the key areas of research in the development of stand-alone transported PDF codes.38... 

At least three approaches have been proposed to solve for the mean pressure field that avoid the noise problem. The first approach is to extract the mean pressure field from a simultaneous consistent39 Reynolds-stress model solved using a standard CFD solver.40 While this approach does alleviate the noise problem, it is intellectually unsatisfying since it leads to a redundancy in the velocity model.41 The second approach seeks to overcome the noise problem by computing the so-called particle-pressure field in an equivalent, but superior, manner (Delarue and Pope 1997). Moreover, this approach leads to a truly... 

Stand-alone refers to the capability of determining the mean pressure field based on the Monte-Carlo simulation without resorting to an external CFD code. 

Unlike (6.178), the Lagrangian model (LGLM) for u has no explicit dependence on the mean pressure field ... 

The first item is easily treated by considering the eigen-modes of the system and expanding the pressure field on a basis formed by these modes. The second item is less well documented but is clearly important. The presence of boundaries not only modifies the structure of the mean flow but also influences the flame dynamics. This is demonstrated in a set of recent experiments in which the lateral confinement was varied systematically [45]. 

As both phases occupy the full flow field concurrently, two sets of conservation equations correspond to these two phases and must be complemented by the set of interfacial jump conditions (discontinuities). A further topological law, relating the void fraction, a, to the phase variables, was needed to compensate for the loss of information due to model simplification (Boure, 1976). One assumption that is often used is the equality of the mean pressures of the two phases, ... 

The source terms on the right-hand sides of Eqs. (25)-(29) are defined as follows. In the momentum balance, g represents gravity and p is the modified pressure. The latter is found by forcing the mean velocity field to be solenoidal (V (U) = 0). In the turbulent-kinetic-energy equation (Eq. 26), Pk is the source term due to mean shear and the final term is dissipation. In the dissipation equation (Eq. 27), the source terms are closures developed on the basis of the form of the turbulent energy spectrum (Pope, 2000). Finally, the source terms... 

It has been shown recently [25] that concentrations of NOj, tend to reduce with increase in the amplitude of discrete-frequency oscillations. The mechanisms remain uncertain, but may be associated with the imposition of a near-sine wave on a skewed Gaussian distribution with consequent reduction in the residence time at the adiabatic flame temperature. Profiles of NO, concentrations in the exit plane of the burner are shown in Fig. 19.6 as a function of the amplitude of oscillations with active control used to regulate the amplitude of pressure oscillations. At an overall equivalence ratio of 0.7, the reduction in the antinodal RMS pressure fluctuation by 12 dB, from around 4 kPa to 1 kPa by the oscillation of fuel in the pilot stream, led to an increase of around 5% in the spatial mean value of NO, compared with a difference of the order of 20% with control by the oscillation of the pressure field in the experiments of [25]. The smaller net increase in NO, emissions in the present flow may be attributed to an increase in NOj due to the reduction in pressure fluctuations that is partly offset by a decrease in NOj, due to the oscillation of fuel on either side of stoichiometry at the centre of the duct. 

This means that, to leading order, there are no spatial variations in the pressure field, although the leading-order pressure may very well remain a function of time. 

From the theory of the stress field around the indenter it follows that almost two thirds of the mean pressure is in the form of a hydrostatic component and therefore plays no part in producing plastic flow. Thus as an approximation... 

The equations for the mean-velocity field Ui and pressure P in an incompressible fluid with constant density and viscosity are... 

FIGURE 10.7 Mean anomaly in sea-level pressure fields from August to October during both inflow seasons without (left) and with MBIs (right) (contour interval 0.2 hPa) (Schinke and Matthaus, 1998). 

For pressure-based techniques, the lack of an independent equation for the pressure complicates the solution of the momentum equation. Furthermore, the continuity equation does not have a transient term in incompressible flows because the fluid transport properties are constant. The continuity reduces to a kinematic constraint on the velocity held. One possible approach is to construct the pressure field so as to guarantee satisfaction of the continuity equation. In this case, the momentum equation still determines the respective velocity components. A frequently used method to obtain an equation for the pressure is based on combining the two equations. This means that the continuity equation, which does not contain the pressure, is employed to determine the pressure. If we take the divergence of the momentum equation, the continuity equation can be used to simplify the resulting equation. 

The components of the momentum equation are usually solved sequentially, meaning that the components of the momentum equation are solved one by one. Since the pressure used in these iterations has been obtained from the previous outer iteration or time step, the velocities computed from (12.159) do not generally satisfy the discretized continuity equation. The predicted velocities do not satisfy the continuity equation, so the uf at iteration 1/ are not the final values of the velocity components. To enforce the continuity equation, the velocities need to be corrected. This is achieved by modifying the pressure field. 

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