State turbulent

In turbulent flow there is a complex interconnected series of circulating or eddy currents in the fluid, generally increasing in scale and intensity with increase of distance from any boundary surface. If, for steady-state turbulent flow, the velocity is measured at any fixed point in the fluid, both its magnitude and direction will be found to vary in a random manner with time. This is because a random velocity component, attributable to the circulation of the fluid in the eddies, is superimposed on the steady state mean velocity. No net motion arises from the eddies and therefore their time average in any direction must be zero. The instantaneous magnitude and direction of velocity at any point is therefore the vector sum of the steady and fluctuating components. 

In general, only turbulent-turbulent flows are of pragmatic interest. The basic Regime I mass balances for the steady state turbulent-turbulent case are... 

APf/L for a liquid in steady state turbulent flow in a rough cylindrical pipe in terms of the liquid density p, the mean velocity u, the inside pipe diameter d, and the roughness e. (b) Use this to calculate AP//L for... 

Calculate the frictional pressure gradient ApyL for a liquid in steady state turbulent flow in a coil of inside tube diameter d,= 0.Q2 m and coil diameter Dc = 2 m if the liquid density... 

In the case of steady-state, turbulent, stratified flows with vanishing pressure gradients according to (3.134) we obtain... 

A better analytically based equation which is valid over a wide range of Prandtl or Schmidt numbers is obtained if we presume a turbulent parallel flow, i.e. a steady-state turbulent flow with vanishing pressure gradient, and velocity, temperature and concentration profiles which are only dependent on the coordinate y normal to the wall. Then, as follows from (3.134) to (3.139),... 

In particular cases simplified reactor models can be obtained neglecting the insignificant terms in the governing microscopic equations (without averaging in space) [9]. For axisymmetrical tubular reactors, the species mass and heat balances are written in cylindrical coordinates. Himelblau and Bischoff [9] give a list of simplified models that might be used to describe tubular reactors with steady-state turbulent flow. A representative model, with radially variable velocity profile, and axial- and radial dispersion coefficients, is given below ... 

With terms III-V in Eq. 4.7 deleted, Eqs. 4.5-4.7, together with the x and y momentum equations, constitute the simplified equations of motion appropriate to natural convection problems. For constant T . and 7U, the boundary conditions on these equations are 0 = 1 and u = v = w = 0 on the body and 0 = 0 far from the body. Steady-state laminar solutions to these equations are those that are obtained after setting the time partials (i.e., terms containing partial derivatives with respect to t ) in the equations equal to zero. Steady-state turbulent... 

This maximum steady state turbulent flame speed depends also on the H2 concentration itself and ranges from about 100 m/sec 10% H2) to about 1800 m/sec (30% b H2 45% H2). Upon exiting the 3 m length of the obstacle (i.e., the Shchelkhin spiral) into the "smooth tube, the flame immediately decelerates. For H2 15%, the flame decelerates to a new and much lower steady state value corresponding to the smooth wall boundary condition. However, for H2 15%, the flame reaccelerates and transits to detonation after a couple of meters of flame travel (the transition distance depends on the H2 concentration). We shall elaborate more on the transition phenomenon in a later section. 

The maximum steady-state turbulent flame speeds as a function... 

Numerous attempts [Dodge and Metzner, 1959 Bogue and Metzner, 1963 Wilson and Thomas, 1985 Shenoy and Talathi, 1985 Shenoy, 1988] have been made at developing analogous expressions for velocity profiles for the steady state turbulent flow of power-law fluids in smoodi pipes most workers have modified the definitions of y+ and but Brodkey et al. [1961] used a polynomial approximation for the velocity profile. Figme 3.13 shows the velocity profiles derived on this basis for power-law fluids the transition region, shown as dotted lines, is least understood. 

For steady-state turbulence, the average value and the rms value of the turbulent fluctuations in a quantity cp can be estimated using time averages computed on a... 

The steady-state nature of the turbulence can be verified by checking that the integrals in [8.5] do not depend on time t from which they are calculated. The recording duration of T should be sufficiently long for the calculated average value to be independent from T. From an experimental standpoint, studying steady-state turbulence constitutes a dramatic simplification. All that is needed is to cany out one experiment of sufficiently long duration. 

The first form relates to the notion of ensemble average, while the second refers, for steadi-state turbulence, to the time average. The following properties result from the definition of probabihties ... 

For steady-state turbulence, the mean energy spectrum is obtained by time averaging over a time interval of duration T ... 

Let us first consider the case of steady-state turbulence (the case of unsustained turbulence is discussed later). An energy source has to inject some energy at a rate equal to the rate s of turbulent kinetic energy dissipation. That is what ideally occurs in a perfectly stirred reactor of volume V (Figure 11.3) in which a moving... 

S.3 Turbulence. As discussed in Section 5-2, there are several steady-state turbulence models in widespread use today. These so-called RANS models address a time-averaged state of the fluid such that all turbulent fluctuations are represented by averaged values. The RANS models are often used with both the MRF and sliding mesh models, as well as with many other transient models used in CFD analysis. This practice is justified in part because the time scales of turbulence fluctuations are assumed small compared to those of the other processes being modeled, such as the blade passing time in a stirred tank. It has also been justified because until recently, other more rigorous treatments have not been available in commercial software or solvable in a realistic time on the computers of the day. 

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