Turbulent flow continuity equation

The basic procedure for the power law fluid is the same as above for the Newtonian fluid. We get a first estimate for the Reynolds number by ignoring fittings and assuming turbulent flow. This is used to estimate the value of / (hence Kpipe) using Eq. (6-44) and the Knt values from the equivalent 3-K equation. Inserting these into Eq. (7-50) then gives a first estimate for the diameter, which is then used to revise the Reynolds number. The iteration continues until successive values agree, as follows ... 

These convective transport equations for heat and species have a similar structure as the NS equations and therefore can easily be solved by the same solver simultaneously with the velocity field. As a matter of fact, they are much simpler to solve than the NS equations since they are linear and do not involve the solution of a pressure term via the continuity equation. In addition, the usual assumption is that spatial or temporal variations in species concentration and temperature do not affect the turbulent-flow field (another example of oneway coupling). 

Note that 7Zu = 0 due to the continuity equation. Thus, the pressure-rate-of-strain tensor s role in a turbulent flow is to redistribute turbulent kinetic energy among the various components of the Reynolds stress tensor. The pressure-diffusion term T is defined... 

In addition to phase change and pyrolysis, mixing between fuel and oxidizer by turbulent motion and molecular diffusion is required to sustain continuous combustion. Turbulence and chemistry interaction is a key issue in virtually all practical combustion processes. The modeling and computational issues involved in these aspects have been covered well in the literature [15, 20-22]. An important factor in the selection of sub-models is computational tractability, which means that the differential or other equations needed to describe a submodel should not be so computationally intensive as to preclude their practical application in three-dimensional Navier-Stokes calculations. In virtually all practical flow field calculations, engineering approximations are required to make the computation tractable. 

It has already been pointed out that the equations of change are valid for describing turbulent flow. The diffusion of A in a nonreacting binary mixture is described by the equation of continuity ... 

For simplicity, consider the steady, incompressible, and isothermal turbulent flows where 9/91 = 0 p = 0 and p = 0. The continuity equation is thus simplified from Eq. (5.58) to... 

In order for a model to be closured, the total number of independent equations has to match the total number of independent variables. For a single-phase flow, the typical independent equations include the continuity equation, momentum equation, energy equation, equation of state for compressible flow, equations for turbulence characteristics in turbulent flows, and relations for laminar transport coefficients (e.g., fJL = f(T)). The typical independent variables may include density, pressure, velocity, temperature, turbulence characteristics, and some laminar transport coefficients. Since the velocity of gas is a vector, the number of independent variables associated with the velocity depends on the number of components of the velocity in question. Similar consideration is also applied to the momentum equation, which is normally written in a vectorial form. 

As an example, for steady, incompressible, and isothermal turbulent flows using the k- model, the independent equations are (1) the continuity equation, Eq. (5.61) (2) the momentum equation, Eq. (5.65) (3) the definition of the effective viscosity, /xeff (combination of Eq. (5.64) and Eq. (5.72)) (4) the equation of turbulent kinetic energy, Eq. (5.75) and (5) the equation for the dissipation rate of turbulent kinetic energy, Eq. (5.80). Thus, for a three-dimensional model, the total number of independent equations is seven. The corresponding independent variables are (1) velocity (three components) (2) pressure (3) effective viscosity (4) turbulent kinetic energy and (5) dissipation rate of turbulent kinetic energy. Thus, the total number of independent variables is also seven, and the model becomes solvable. 

This is the continuity equation for turbulent flow when the mean motion is two-dimensional. It will be noted that this equation has exactly the same form as the continuity equation for two-dimensional steady laminar flow with the mean values of the velocity components substituted in place of the steady values that apply in laminar flow. This result can, in fact, be deduced by intuitive reasoning and simply states that if an elemental control volume through which the fluid flows is considered, then over a sufficiently long period of time, the fluctuating components contribute nothing to the mass transfer through this control volume. 

Now if the continuity equation is multiplied by u and then applied to turbulent flow the following is obtained... 

In many cases, a region of laniinar flow will have to be allowed for over the initial part of the body. The calculation bf this portion of the flow can be accomplished by using the same equations as for turbulent flow with E set equal to 0. This calculation can be started using initial conditions of the type discussed in Chapter 3 for purely laminar flows. From the point where transition is assumed to occur, the calculation can be continued using the equations presented above to describe E. 

The modelling of super critical water oxidation (SCWO), up to now not been used in large scale industrial applications, is important for design of pilot plants and, later, industrial plants. The applied programme to model the continuous flow in a reactor is called CAST (Computer Aided Simulation of Turbulent Flows [8]) and is based on the method of the finite volume. That means that the balance equations were integrated over the surfaces of each control volume. 

Consider the momentum change that occurs when a fluid flows steadily through the pipe-work shown in Figure 1.3. It will be assumed that the axial velocity component is uniform over the cross section and equal to u. This is a good approximation for turbulent flow. The x-momentum flow rate into the section across plane 1 is equal to AfiUt and that out of the section across plane 2 is equal to M2U2. By continuity, Mx = M2 = M. From equation 1.20, the rate of change of momentum is given by... 

In Section 4.2.4, the governing equations of fluid mechanics for a turbulent flow are derived. Similarly, the governing equations for heat transfer and mass transfer can be derived from the principles of energy and mass conservation. In fact, the species conservation equation is an extension of the overall mass conservation (or the continuity) equation. For species i, it has the following form ... 

For definiteness, we consider the transfer processes between a cylindrical wall and a turbulently flowing n-component fluid mixture. For condensation of vapor mixtures flowing inside a vertical tube, for example, the wall can be considered to be the surface of the liquid condensate film. We examine the phenomena occurring at any axial position in the tube, assuming that fully developed flow conditions are attained. For steady-state conditions, the equations of continuity of mass of component i (assuming no chemical reactions), Eqs. 1.3.7 take the form... 

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