Mean velocity field

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... 

Thus, the mean velocity field - as well as the fluctuation field u - are solenoidal. 

In summary, the mean velocity field (U) could be found by solving (2.93) and (2.98) if a closure were available for the Reynolds stresses. Thus, we next derive the transport equation for lutu ) starting from the momentum equation. 

Applying the continuity equation for the mean velocity field, Ce is exactly zero at high Reynolds numbers. 

The key step here is to use the conditional PDF to eliminate the velocity dependence. However, this generates a new unclosed term. Note that we assume the mean velocity field to be solenoidal in the last line. 

Finally, using the fact that the mean velocity field is solenoidal, the RANS mean velocity transport equation reduces to... 

However, the mean velocity field will depend on conditional moments of the fluctuating velocity through the mean velocity transport equation,... 

For multi-dimensional flows, the mean velocity field may be a function of x. For this case, deterministic errors can be introduced through (V )(A-I" t), resulting in a non-uniform distribution for Xin, (t). This is an important source of numerical error in transported PDF codes, and we will look at this problem in more detail in Chapter 7. 

The random selection in step (iii) is carried out by generating uniform random numbers U e [0, 1], For example, the index of a random particle selected from a set of N particles will be n = intup(//N) where intuP() rounds the argument up to the nearest integer. Note that for constant-density, statistically stationary flow, the effective flow rates will be constant. In this case, steps (i) and (ii) must be completed only once, and the MC simulation is advanced in time by repeating step (iii) and intra-cell processes. For variable-density flow, the mean density field ((p)) must be estimated from the notional particles and passed back to the FV code. In the FV code, the non-uniform density field is held constant when solving for the mean velocity field.15... 

Strictly speaking, this can only be true if the interpolated mean velocity field satisfies continuity. 

In the presence of a mean scalar gradient V and a fluctuating (zero-mean) velocity field... 

The mean velocity field is obtained by averaging 60 instantaneous velocity fields. For these conditions the flame is anchored to the collar lip. Superimposed on... 

The mean velocity field is found to be self-similar for all the nozzle exit velocities, distances between nozzles and turbulence generators tested. This similarity allows the mean axial velocity traverses to be normalized so that all the measured data lie on the same curve. The shape of the curve is similar to that given in Fig. 1.8. 

It is usually necessary to match the refractive indices of two fluids (and the transparent wall of flow passage in some cases particularly for microchannel flow). For example, in an experimental study on the selfpreserving structure of steady round buoyant turbulent plums in cross flow (Diez et al., 2005), planar-LIF (PLIF) and PIV techniques are utilized to measure the mean concentration of source fluid and mean velocity fields simultaneously. Both PLIF and PIV measurements in this study necessitate matching the indices of refraction of the source (water solution of potassium phosphate, monobasic KH2PO4, containing Rhodamine 6G dye) and ambient fluids (ethyl alcohol/water) to avoid scattering the laser beam away from the buoyant flow. Visual inspection... 

The smallness of the perturbation quantities are indicated by the small parameter e. Furthermore, the mean velocity field is assumed parallel/ quasi-parallel so that,... 

The acronyms for closure type used in this review are as follows FVF, fluctuating velocity field MVF, mean-velocity field MVFN, Newtonian MVF MTE, mean turbulent energy MTEN, Newtonian MTE MTOS, structural MTE MTEN/L, MTEN closure with dynamical length scale equation MRS, mean Reynolds-stress MRS/L, MRS closure with dynamical length scale equation. 

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

Closure is obtained through assumptions that relate the Reynolds stresses Rii to properties of the mean velocity field Ui. The most productive approach has been to use a consitutive equation involving a turbulence length scale, usually called the mixing length. A generalization of the... 

Gawain and Pritchard (Gl) proposed a more complicated hueristic integrodifferential equation for turbulent length scales. In effect their local length scale is determined by the mean velocity field in the region of the local point. The two-point tensor... 

The ability of MTEN calculations to predict accurately the mean velocity field and turbulence kinetic-energy distribution is demonstrated by Fig. 17 from Mellor and Herring s contribution to the Stanford conference. Their use of a fine computational mesh near the wall is reflected in their accurate prediction of the inner regions. 

FIGURE 10.8 Comparison of predicted mean velocity field with angle-averaged PIV data (impeller center plane) (from Ranade et al., 2001 b). (a) Radial mean velocity (b) Tangential mean velocity. 

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