Stresses, Reynolds

The diagonal components of the Reynolds stress tensor (e.g., (u, U )) are referred to as the Reynolds normal stresses, while the off-diagonal components are referred to as the 

A governing equation for the mean pressure field appearing in (2.93) can be found by Reynolds averaging (1.29). This leads to a Poisson equation of the form 

As it stands, the last term on the right-hand side of this expression is non-linear in the spatial derivatives and appears to add a new closure problem. However, using the fact that the fluctuation field is 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 (UiUj) starting from the momentum equation. 

The transport equation for the Reynolds stresses can be found starting from the governing equation for the velocity fluctuations  

In Sec. 6.4 we discussed how turbulent fluctuations lead to shear stresses in addition to those due to simple viscous shear. These additional stresses are called Reynolds stresses. It can be shown mathematically [9, p. 559] that the Reynolds stress components for a general three-dimensional flow (see Sec. 7.7) are given by 

The most interesting of the Reynolds stresses are the shear stresses. From Eq. 16.14 we see that these require the fluctuations to be in two directions. If 

Now some time later a mass of fluid from C is brought to B by an eddy. By similar arguments, Vy is negative and is positive. Thus, for both kinds of fluctuations is negative, and there is, indeed, a correlation between the two velocity fluctuations at right angles to each other and, hence, a significant Reynolds stress. 

The magnitude and character of these Reynolds stresses can be visualized through the concept of the eddy viscosity, first introduced by Boussinesq [1, p. 23]. He suggested that we retain the form of Newton s law of viscosity 

Diagram showing how Reynolds stresses arise in a turbulent flow with a velocity gradient. 

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Reynolds number Re Reynolds numbers Reynolds stresses Rezipas... 

Reynolds Stress Models. Eddy viscosity is a useful concept from a computational perspective, but it has questionable physical basis. Models employing eddy viscosity assume that the turbulence is isotropic, ie, u u = u u = and u[ u = u u = u[ = 0. Another limitation is that the... 

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 ... 

The Reynolds stresses are nonzero because the velocity fluctuations in different coordinate directions are correlated so that in general is nonzero. 

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... 

These extra turbulent stresses are termed the Reynolds stresses. In turbulent flows, the normal stresses -pu, -pv, and -pw are always non-zero beeause they eontain squared veloeity fluetuations. The shear stresses -pu v, -pu w, -pv w and are assoeiated with eorrelations between different veloeity eomponents. If, for instanee, u and v were statistieally independent fluetuations, the time average of their produet u v would be zero. However, the turbulent stresses are also non-zero and are usually large eompared to the viseous stresses in a turbulent flow. Equations 10-22 to 10-24 are known as the Reynolds equations. 

The flow pattern is ealeulated from eonservation equations for mass and mometum, in eombination with the Algebraie Stress Model (ASM) for the turbulent Reynolds stresses, using the Fluent V3.03 solver. These equations ean be found in numerous textbooks and will not be reiterated here. Onee the flow pattern is known, the mixing and transport of ehemieal speeies ean be ealeulated from the following model equation ... 

Almost all modern CFD codes have a k - model. Advanced models like algebraic stress models or Reynolds stress model are provided FLUENT, PHOENICS and FLOW3D. Table 10-3 summarizes the capabilities of some widely used commercial CFD codes. Other commercially CFD codes can be readily assessed on the web from hptt//www.cfd-online.com This is largest CFD site on the net that provides various facilities such as a comprehensive link section and discussion forum. 

Turbulence modeling capability (range of models). Eddy viscosity k-1, k-e, and Reynolds stress. k-e and Algebraic stress. Reynolds stress and renormalization group theory (RNG) V. 4.2 k-e. low Reynolds No.. Algebraic stress. Reynolds stress and Reynolds flux. k- Mixing length (user subroutine) and k-e. 

Chen, Q. Prediction of room air motion by Reynolds-stress models. Build. FInviron., vol. 31, pp. 233-244, 1996. 

Rodi, W. A new algebraic relation for calculating the Reynolds stresse.s. ZAMM, vol. 56, pp. T219-T221, 1976. 

An appropriate model of the Reynolds stress tensor is vital for an accurate prediction of the fluid flow in cyclones, and this also affects the particle flow simulations. This is because the highly rotating fluid flow produces a. strong nonisotropy in the turbulent structure that causes some of the most popular turbulence models, such as the standard k-e turbulence model, to produce inaccurate predictions of the fluid flow. The Reynolds stress models (RSMs) perform much better, but one of the major drawbacks of these methods is their very complex formulation, which often makes it difficult to both implement the method and obtain convergence. The renormalization group (RNG) turbulence model has been employed by some researchers for the fluid flow in cyclones, and some reasonably good predictions have been obtained for the fluid flow. 

Consequently, six additional unknowns, the Reynolds stresses obtained and the equations for turbulent flow beeome... 

Using turbulenee models, this new system of equations ean be elosed. The most widely used turbulenee model is the k-e model, whieh is based on an analogy of viseous and Reynolds stresses. Two additional transport equations for the turbulent kinetie energy k and the turbulent energy dissipation e deseribe the influenee of turbulenee... 

More advanced models, for example the algebraic stress model (ASM) and the Reynolds stress model (RSM), are not based on the eddy-viscosity concept and can thus account for anisotropic turbulence thereby giving still better predictions of flows. In addition to the transport equations, however, the algebraic equations for the Reynolds stress tensor also have to be solved. These models are therefore computationally far more complex than simple closure models (Kuipers and van Swaaij, 1997). 

In equations 12.19 and 12.20, Ry represents the momentum transferred per unit area and unit time. This momentum transfer tends to accelerate the slower moving fluid close to the surface and to retard the faster-moving fluid situated at a distance from the surface. It gives rise to a stress Ry at a distance y from the surface since, from Newton s Law of Motion, force equals rate of change of momentum. Such stresses, caused by the random motion in the eddies, are sometimes referred to as Reynolds Stresses. 

DNS results are usually considered as references providing the same level of accuracy as experimental data. The maximum attainable Reynolds number (Re) in a DNS is, however, too low to duplicate most practical turbulent reacting flows, and hence, the use of DNS is neither to replace experiments nor for direct comparisons— not yet at least. However, DNS results can be used to investigate three-dimensional (3D) features of the flow (coherent structures, Reynolds stresses, etc.) that are extremely difficult, and sometimes impossible, to measure. One example of such achievement for nonreacting... 

Tahry, S.E., Application of a Reynolds stress model to engine-like flow calculations. /. of Fluids Engineering, 1985.107(4) 444-450. 

Ekambra etal. [21] compared the results from ID, 2D, and 3D simulations of a bubble column with experimental results. They obtained similar results for holdup and axial velocity, while eddy viscosity, Reynolds stresses, and energy dissipation were very different in the three simulations as shown in Figure 15.7. This example also illustrates the importance of selecting the right variables for model vahdation. A 2D model will yield good results for velocity but will predict all variables based on turbulent characteristics poorly. 

Figure 15.7 Measured and simulated holdup (a), axial velocity (b), eddy viscosity (c), and Reynolds stresses (d), using a ID, 2D, and 3D simulations (From [21]).

In the velocity field of the determining eddies, which is characterized by the turbulent fluctuation velocity the particles experience a dynamic stress according to the Reynolds stress Eq. (2) ... 

To avoid gas-liquid mass transfer Hmitation, which would have a negative influence on productivity, in correctly operated bioreactors there are turbulent flow conditions with more or less pronounced turbulence, for which the Reynolds stress formula (Eq. (2)) can be used. Whereas, as a rule there is fully developed turbulent flow in technical apparatuses (see condition (6) and explanations in Sect. 8), this is frequently not the case in laboratory fermenters. Equations (3) and (4) are then only valid to a limited extent. 

Where the Reynolds stress formula (2) and the universal law of the theory of isotropic turbulence apply to the turbulent velocity fluctuations (4), the relationship (20) for the description of the maximum energy dissipation can be derived from the correlation of the particle diameter (see Fig. 9). It includes the geometrical function F and thus provides a detailed description of the stirrer geometry in the investigated range of impeller and reactor geometry 0.225derived from many turbulence measurements, correlation (9). 

Hydrodynamic effects on suspended particles in an STR may be broadly categorized as time-averaged, time-dependent and collision-related. Time-averaged shear rates are most commonly considered. Maximum shear rates, and accordingly maximum stresses, are assumed to occur in the impeller region. Time-dependent effects, on the other hand, are attributable to turbulent velocity fluctuations. The relevant turbulent Reynolds stresses are frequently evaluated in terms of the characteristic size and velocity of the turbulent eddies and are generally found to predominate over viscous effects. 

For certain conditions, the hydrodynamic stresses generated by the very rapid fluctuations in turbulent flow, the so-called Reynolds stresses, can be estimated as [70] ... 

For the same bioreactor mentioned above (operating at an impeller tip speed of 1.4 ms ), Dunlop et al. [57] predicted maximum Reynolds stresses in the impeller region of 32.4 Nm. However, if the energy is assumed to be uniformly dissipated throughout the vessel contents, then Eq. (5) will yield lower values. As calculated, the Reynolds stresses involve a length scale and the stress experienced by a particular entity will depend on its size. 

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