Incompressible flow

The quantity dq s normally taken as the specific external heating to the differential element, and the quantity dw as the specific useful work abstracted. However, the relationship defined by equation (4.7) applies equally when the specific heat input and the specific work done are the total specific heat input and the total specific work, where the latter include frictional heating and work, as we shall now show. 

In a frictionally resisted flow, the total work done will be the sum of the useful work, dw, and the work done against friction, dF, per unit mass of fluid, in units of Joules per kilogram  

But the frictional work will reappear as heat, so the total heating per unit mass is now the sum of the externally imposed heating and the frictional heating  

Substituting from equation (4.8) and (4.9) back into equation (4.7) gives 

Notice that equations (4.7) and (4.10) have the same form, except that dq and dw have been replaced by dq, and dw, in the latter. We will now develop equation (4.10) for the case of pipe flow. 

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In this chapter the general equations of laminar, non-Newtonian, non-isothermal, incompressible flow, commonly used to model polymer processing operations, are presented. Throughout this chapter, for the simplicity of presentation, vector notations are used and all of the equations are given in a fixed (stationary or Eulerian) coordinate system. 

Weighted residual finite element methods described in Chapter 2 provide effective solution schemes for incompressible flow problems. The main characteristics of these schemes and their application to polymer flow models are described in the present chapter. 

U-V-P schemes belong to the general category of mixed finite element techniques (Zienkiewicz and Taylor, 1994). In these techniques both velocity and pressure in the governing equations of incompressible flow are regarded as primitive variables and are discretized as unknowns. The method is named after its most commonly used two-dimensional Cartesian version in which U, V and P represent velocity components and pressure, respectively. To describe this scheme we consider the governing equations of incompressible non-Newtonian flow (Equations (1.1) and (1.4), Chapter 1) expressed as... 

Further details of the BB, sometimes referred to as Ladyzhenskaya-Babuska-Brezi (LBB) condition and its importance in the numerical solution of incompressible flow equations can be found in textbooks dealing with the theoretical aspects of the finite element method (e.g. see Reddy, 1986), In practice, the instability (or checker-boarding) of pressure in the U-V-P method can be avoided using a variety of strategies. 

The main strategies for obtaining stable results by the U -V -P scheme for incompressible flow are as follows ... 

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). 

As already explained the necessity to satisfy the BB stability condition restricts the types of available elements in the modelling of incompressible flow problems by the U-V P method. To eliminate this restriction the continuity equation representing the incompressible flow is replaced by an equation corresponding to slightly compressible fluids, given as... 

Using different types of time-stepping techniques Zienkiewicz and Wu (1991) showed that equation set (3.5) generates naturally stable schemes for incompressible flows. This resolves the problem of mixed interpolation in the U-V-P formulations and schemes that utilise equal order shape functions for pressure and velocity components can be developed. Steady-state solutions are also obtainable from this scheme using iteration cycles. This may, however, increase computational cost of the solutions in comparison to direct simulation of steady-state problems. 

Elimination of the pressure term from the equation of motion does not automatically yield a robust scheme for incompressible flow and it is still necessary to satisfy the BB stability condition by a suitable technique in both forms of the penalty method. 

The use of selectively reduced integration to obtain accurate non-trivial solutions for incompressible flow problems by the continuous penalty method is not robust and failure may occur. An alternative method called the discrete penalty technique was therefore developed. In this technique separate discretizations for the equation of motion and the penalty relationship (3.6) are first obtained and then the pressure in the equation of motion is substituted using these discretized forms. Finite elements used in conjunction with the discrete penalty scheme must provide appropriate interpolation orders for velocity and pressure to satisfy the BB condition. This is in contrast to the continuous penalty method in which the satisfaction of the stability condition is achieved indirectly through... 

The basic procedure for the derivation of a least squares finite element scheme is described in Chapter 2, Section 2.4. Using this procedure the working equations of the least-squares finite element scheme for an incompressible flow are derived as follows ... 

Kheshgi, H. S. and Scriven, L. E., 1985. Variable penalty method for finite element analysis of incompressible flow. Int. J. Numer. Methods Fluids 5, 785-803. 

Tanner, R.I. 2000. Engineering Rheology, 2nd edti, Oxford University Press, Oxford. Taylor, C., Ranee, J. and Medwell, J. O., 1985. A note on the imposition of traction boundary conditions when using FEM for solving incompressible flow problems. Comnmn. Appl. Numer. Methods 1, 113-121. 

Equations 34 and 36 are appHed to equations 1—6. After time-averaging, the following equations are obtained for incompressible flows ... 

The principle can be illustrated by examining the Navier-Stokes equation for two-dimensional incompressible flow. The x-component of the equation is... 

Landau instabilities are the hydrodynamic instabilities of flame sheets that are associated neither with acoustics nor with buoyancy but instead involve only the density decrease produced by combustion in incompressible flow. The mechanism of Landau instability is purely hydrodynamic. In principle, Landau instabilities should always be present in premixed flames, but in practice they are seldom observed (26,27). 

Vorticity The relative motion between two points in a fluid can be decomposed into three components rotation, dilatation, and deformation. The rate of deformation tensor has been defined. Dilatation refers to the volumetric expansion or compression of the fluid, and vanishes for incompressible flow. Rotation is described bv a tensor (Oy = dvj/dxj — dvj/dxi. The vector of vorticity given by one-half the... 

Here 4 = F,Jfn is the energy dissipation per unit mass. This equation has been called the engineering Bernoulli equation. For an incompressible flow, Eq. (6-15) becomes... 

The identity tensor by is zero for i J and unity for i =J. The coefficient X is a material property related to the bulk viscosity, K = X + 2 l/3. There is considerable uncertainty about the value of K. Traditionally, Stokes hypothesis, K = 0, has been invoked, but the vahdity of this hypothesis is doubtful (Slattery, ibid.). For incompressible flow, the value of bulk viscosity is immaterial as Eq. (6-23) reduces to... 

Mechanical Energy Balance The mechanical energy balance, Eq. (6-16), for fully developed incompressible flow in a straight circular pipe of constant diameter D reduces to... 

Vfjp is the friction velocity and =/pVV2 is the wall stress. The friction velocity is of the order of the root mean square velocity fluctuation perpendicular to the wall in the turbulent core. The dimensionless distance from the wall is y+ = yu p/. . The universal velocity profile is vahd in the wall region for any cross-sectional channel shape. For incompressible flow in constant diameter circular pipes, = AP/4L where AP is the pressure drop in length L. In circular pipes, Eq. (6-44) gives a surprisingly good fit to experimental results over the entire cross section of the pipe, even though it is based on assumptions which are vahd only near the pipe wall. 

Losses 4 fof incompressible flow in sections of straight pipe of constant diameter may be calculated as previously described using the Fanning fric tion fac tor ... 

Equation (6-95) is valid for incompressible flow. For compressible flows, see Benedict, Wyler, Dudek, and Gleed (J. E/ig. Power, 98, 327-334 [1976]). For an infinite expansion, A1/A2 = 0, Eq. (6-95) shows that the exit loss from a pipe is 1 velocity head. This result is easily deduced from the mechanic energy balance Eq. (6-90), noting that Pi =pg. This exit loss is due to the dissipation of the discharged jet there is no pressure drop at the exit. 

Screens The pressure drop for incompressible flow across a screen of fractionaf free area Ot may be computed from... 

Section 10 of this Handbook describes the use of orifice meters for flow measurement. In addition, orifices are commonly found within pipelines as flow-restric ting devices, in perforated pipe distributing and return manifolds, and in perforated plates. Incompressible flow through an orifice in a pipehne as shown in Fig. 6-18, is commonly described by the following equation for flow rate Q in terms of pressure drop across the orifice Ap, the orifice area A, the pipe cross-sectional area A, and the density p. 

Porous Media Packed beds of granular solids are one type of the general class referred to as porous media, which include geological formations such as petroleum reservoirs and aquifers, manufactured materials such as sintered metals and porous catalysts, burning coal or char particles, and textile fabrics, to name a few. Pressure drop for incompressible flow across a porous medium has the same quahtative behavior as that given by Leva s correlation in the preceding. At low Reynolds numbers, viscous forces dominate and pressure drop is proportional to fluid viscosity and superficial velocity, and at high Reynolds numbers, pressure drop is proportional to fluid density and to the square of superficial velocity. 

Many transient flows of liquids may be analyzed by using the full time-dependent equations of motion for incompressible flow. However, there are some phenomena that are controlled by the small compressibility of liquids. These phenomena are generally called hydraulic transients. 

When the continmty equation and the Navier-Stokes equations for incompressible flow are time averaged, equations for the time-averaged velocities and pressures are obtained which appear identical to the original equations (6-18 through 6-28), except for the appearance of additional terms in the Navier-Stokes equations. Called Reynolds stress terms, they result from the nonlinear effects of momentum transport by the velocity fluctuations. In each i-component (i = X, y, z) Navier-Stokes equation, the following additional terms appear on the right-hand side ... 

Closure Models Many closure models have been proposed. A few of the more important ones are introduced here. Many employ the Boussinesq approximation, simphfied here for incompressible flow, which treats the Reynolds stresses as analogous to viscous stresses, introducing a scalar quantity called the turbulent or eddy viscosity... 

This chapter is a brief diseussion of large deformation wave codes for multiple material problems and their applications. There are numerous other reviews that should be studied [7], [8]. There are reviews on transient dynamics codes for modeling gas flow over an airfoil, incompressible flow, electromagnetism, shock modeling in a single fluid, and other types of transient problems not addressed in this chapter. 

For any control valve design be sure to use one of the modem methods, such as that given here, that takes into account such things as control valve pressure recovery factors and gas transition to incompressible flow at critical pressure drop. 

Parameters specified are mass flow or velocity. Usually at one outlet, pressure equal to a constant is specified in incompressible flow. If several outlets are present, this pressure boundary condition can only be applied to one outlet, as there are some (unknown) pressure differences between the different outlets. The flow conditions in the rooms are better represented by taking the outlet mass flows when they are known. 

The continuity equation for an incompressible flow is given by the following expression ... 

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