Algorithmically incompressible

Algorithms based on the last approach usually provide more flexible schemes than the other two methods and hence are briefly discussed in here. Hughes et al. (1986) and de Sampaio (1991) developed Petrov-Galerkin schemes based on equal order interpolations of field variables that used specially modified weight functions to generate stable finite element computations in incompressible flow. These schemes are shown to be the special cases of the method described in the following section developed by Zienkiewicz and Wu (1991). 

CFD methods are used for incompressible- and compressible-, creeping-, laminar- and turbulent-, Newtonian- and non-Newtonian-, and isothermal- and non-isothermal flows. Most commercial CFD codes include the k-z turbulence model [10]. More accurate models are also becoming available. The accuracy of the solution depends on how the mesh fits the true geometry, on the convergence of the solution algorithm, and also on the model used to describe the turbulent flow [11]. 

P 61] The numerical simulations were based on the solution of the incompressible Navier-Stokes equation and a convection-diffusion equation for a concentration field by means of the finite-volume method [152], The Einstein convention of summation over repeated indices was used. For pressure-velocity coupling, the SIMPLEC algorithm and for discretization of the species concentration equation the QUICK differencing scheme were applied. Hybrid and the central differencing schemes referred to velocities and pressure, respectively (commercial flow solvers CFX4 and CFX5). 

This is the most commonly used model for natural gas nets, and most algorithms for incompressible networks may be used for gas networks as well, simply by replacing eqn (3) by eqn (4) in the library of pressure drop correlations. There are several commercially available computer programs for gas networks, among which the ones from the British Gas Corporation (6) and Intercomp (7) are found. 

Consider flow over a flat plate. The computation is started by assuming that uy = u,y, at the leading edge and v, = 0. The value of vy is needed in the explicit algorithm to move on to the i+1 level. It is not required to specify the initial values of vy in the formal mathematical formulation of the partial differential equation. A suitable initial distribution for vy can be obtained by using the continuity equation to eliminate du/5x from the x-momentum equation. For a laminar, incompressible flow, this means that... 

Equation (10) holds for any function V vanishing on Fi. The last temi of the augmented Lagragian (for r=0, Lr is a Lagrangian) introduces a penalty of the incompressibility condition and the Uzawa algorithm allows us to satisfy equation (3) as precisely as we wish using moderate values of r. 

Leveque, R.J. (1996), High-resolution conservative algorithms foradvection in incompressible flow, SIAM J. Numer. Anal., 33, 627-665. 

Many other methods for solving flow problems can be devised. It is impossible to describe all of them here. In this book, emphasis is placed on describing elements of particular pressure-based methods originally developed for incompressible flows. The basic methods are extended and used to simulate reactive flows. The standard algorithms used to solve multi-fluid models are extensions of particular pressure-based methods for single phase flows. 

Mercle CL, Athavale M (1987) Time-Accurate Unsteady Incompressible Flow Algorithms Based on Artificial Compressibility. AIAA Paper 87-1137, AIAA Press Washington, DC... 

The governing equations (1) and (2) are of a mixed parabolic-elliptic nature. A key feature of incompressible flow is that that the time derivative of pressure vanishes from the equations. Hence the equations do not transmit any pressure history directly, and it is as if a new pressure field is established at each step. This situation does not arise for compressible flow where, owing to the presence of the time derivative of the pressure term in the continuity equation, one can solve the coupled hyperbolic system by advancing in time. In the absence of such a term, the algebraic system of equations becomes singular. This is also why attempts to solve the incompressible flow problem as a low Mach-number, compressible-flow problem lead to ill-conditioned algebraic systems with poor algorithmic efficiency and accuracy. For a detailed discussion of these issues, see Ref. 74, p. 642. 

Lattice-gas algorithms have shown to provide useful information about the dynamics of flow in porous media and particularly in multiple-scaled porous media (Di Pietro, 1996 Di Pietro et al., 1994). These models reproduce fluid behavior within the incompressible limit, because they are automata for the Navier-Stokes equations. Lattice-gases allow the study of hydrodynamic phenomena at the pore scale, but they also recover transport laws at a macroscopic scale (Di Pietro, 1998). 

The two-fluid formulation consists of solving the governing equations in both fluids independently and then matching the interfacial boundary conditions at the interface, which usually requires an iterative algorithm. This approach keeps the interface as a discontinuity, consistent with the continuum mechanics concept. For each phase, we can write the following momentum equation along with the incompressibility constraint ... 

The computation starts with a predictor-corrector algorithm for the determination of velocity field at to + At. In the predictor stage of the solution algorithm, the pressure is replaced by an arbitrary pseudo-pressure P (which in most cases is set equal to zero at full cells), and tentative velocities are then calculated. A pseudo-pressure boundary condition is applied in surface cells to satisfy the normal stress condition. Since pressure has been ignored in the full cells, the tentative velocity field does not satisfy the incompressible continuity equation. The deviation fi om incompressibility is used to calculate a pressure potential field [J/, which then is used to correct the velocity field. In the final steps, the velocity boundary conditions are calculated, the new location of free surface is determined by tracking the markers, and the velocity boundary conditions associated with the new fluid cells are assigned (Fig. 5). 

The main output stages of the two-component viscous incompressible fluid model were considered and numerical algorithm for solving the resulting model was chosen as well in this work. Calcrflations for two-dimensional and three-dimensional problems of the wave emergence and propagation on the free surface were carried out. 

Most soft elastomers are nearly incompressible and the adoption of the hypothesis of incompressibility in the constitutive equations allows for some simplifications in the analytical calculations. However, this choice has some shortcomings (i) die constraint/ = 1 poses difficulties in the convergence of computational algorithms when numerical solutions are sought and (ii) the stmcture of die constitutive electroelastic equations are not completely revealed because of die presence of the hydrostatic pressure. Therefore, aiming at elucidating the latter item, we prefer to formulate the constitutive equations and solve a relevant boundary-value problem, for a soft dielectric beginning from the compressible case. 

---