Turbulent shearing stress

Eigure 20 compares the predictions of the k-Q, RSM, and ASM models and experimental data for the growth of the layer width 5 and the variation of the maximum turbulent kinetic energy k and turbulent shear stress normalized with respect to the friction velocity jp for a curved mixing layer... 

Power is the external measure of the mixer performance. The power put into the system must be absorbed through friction in viscous and turbulent shear stresses and dissipated as heat The power requirement of a system is a function of the impeller shape, size, speed of rotation, fluid density and viscosity, vessel dimensions and internal attachments, and posidon of the impeller in this enclosed system. 

Fig. 15. Effects of turbulent shear stress level and exposure time on cell viability measured by trypan blue staining. Cells were sheared in a concentric cylinder viscometer [1]...

The average t0 of rox over the path length x0 should be considered as the local turbulent shear stress at the wall. Hence... 

A consideration of the right-hand sides of these two equations indicates that the turbulence terms in these equations have, as discussed in Chapter 2, the form of additional shearing stress and heat transfer terms although they arise, of course, from the momentum transfer and enthalpy transfer produced by the mixing that arises from the turbulence. Because of their similarity to the molecular terms, the turbulence terms are usually called the turbulent shear stress and turbulent heat transfer rate respectively. Thus, the following are defined ... 

Tp being the turbulent shear stress and qj being the turbulent heat transfer rate. 

By analogy with the form of the molecular shearing stress and molecular heat transfer rate relations, as given in Eqs. (5.6) and (5.7), it is often convenient to express the turbulent shearing stress and turbulent heat transfer in terms of the velocity and temperature gradients in the following way ... 

In writing these equations, account has been taken of the fact that the sign of tt depends on the sign of duldy. Hence, duldy (dutdy) has been used instead of (Su/dy)2. The above equation for the turbulent shear stress is conveniently written as ... 

In order to utilize this equation to determine the turbulent shear stress it is necessary to obtain an additional equation relating, for example, the eddy viscosity to die quantities involved in the turbulent kinetic energy equation. If it is assumed that ... 

Next consider the inner layer. Because it is being assumed that turbulent shearing stress and heat transfer rate are negligible in this region, the total shearing stress and heat transfer rate in this layer are given by ... 

Many experiments have established that, as mentioned before, there is a region near the wall where the local turbulent shear stress depends on the wall shear stress and the distance from the wall alone and is largely independent of the nature of the rest of the flow. In this region, the mixing length increases linearly with distance from the wall except that near the wall there is a damping of the turbulence due to viscosity. In the wall region, it is assumed therefore that ... 

Solution. In forced convective flow the turbulent shear stress is given by ... 

The fluctuations give rise to a turbulent-shear stress which may be analyzed by referring to Fig. 5-12. 

For a unit area of the plane P-P, the instantaneous turbulent mass-transport rate across the plane is pv. Associated with this mass transport is a change in the x component of velocity u. The net momentum flux per unit area, in the x direction, represents the turbulent-shear stress at the plane P-P, or pv u When a turbulent lump moves upward (v > 0), it enters a region of higher u and is therefore likely to effect a slowing-down fluctuation in u , that is, u < 0. A similar argument can be made for v < 0, so that the average turbulent-shear stress will be given as... 

Let us imagine a turbulent lump which is located a distance above or below the plane P-P. as shown in Fig. 5-12. These lumps of fluid move back and forth across the plane and give rise to the eddy or turbulent-shear-stress effect. At y + ( the velocity would be approximately... 

The distance (. is called the Prandtl mixing length. Prandtl also postulated that v would be of the same order of magnitude as u so that the turbulent-shear stress of Eq. (5-60) could be written... 

Bradshaw et al. (B3) use Eqs. (40) to derive a differential equation for the turbulent shear stress t. The transport velocity Qa is taken as (Tmei/p), where Tm x is the maximum value of riy) in the boundary layer. G and I are prescribed as functions of the position across the boundary layer, and o is essentially taken as constant. Together with Eqs. (10a,b), Eq. (36) gives a closed set of equations for U, V, and t this system is of hyperbolic type, with three real characteristic lines. Bradshaw et al. construct a numerical solution using the method of characteristics it can also be done using small streamwise steps with an explicit difference scheme (Nl A. J. Wheeler and J. P. Johnston, private communications). There is a great physical appeal to the characteristics, especially since it is found that the solutions along the outward-going characteristic dominates the total solution. This... 

Perhaps the first thought that comes to mind is to determine the shear stress in an analogous manner to laminar flow from t - -p, duldr, where u(r) is the average velocity profile for turbulent flow. But the experimental studies show that this is not the case, and the shear stress is much larger due to the turbulent fluctuations. Therefore, it is convenient to think of the turbulent shear stress as consisting of two parts the laininai component, wlticli accounts for... 

In narrow gap the area of conventional nucleate boiling could be observed just in small bubble flow regime. The balance between capillary forces and turbulent shear stress gives the maximum stable diameter of the bubble in turbulent flow as [26]... 

An exclusively analytical treatment of heat and mass transfer in turbulent flow in pipes fails because to date the turbulent shear stress Tl j = —Qw w p heat flux q = —Qcpw, T and also the turbulent diffusional flux j Ai = —gwcannot be investigated in a purely theoretical manner. Rather, we have to rely on experiments. In contrast to laminar flow, turbulent flow in pipes is both hydrodynamically and thermally fully developed after only a short distance x/d > 10 to 60, due to the intensive momentum exchange. This simplifies the representation of the heat and mass transfer coefficients by equations. Simple correlations, which are sufficiently accurate for the description of fully developed turbulent flow, can be found by... 

As shown in Table 2.4, and reviewed by Britter and Hanna 2003 [81], most FAM s for the neighbourhood scale tend to focus on equilibrium flows within and just above the canopy. In a porous canopy (Figure 2.2), the mean velocity within the canopy Uc(z) is driven by the turbulent shear stresses generated in the intense shear layer just above the canopy. Here the ratio of Uc/U(z ), where U(z ) is the velocity at the top of... 

Figure 3.31 Transformation of the mean velocity U(z) (A), its fluctuation U (z) (B), and turbulent shear stress (C) along the D-canopy with the crowns normal to the air flow.

Two traditional approaches to the closure of the Reynolds equation are outlined below. These approaches are based on Boussinesq s model of turbulent viscosity completed by Prandtl s or von Karman s hypotheses [276, 427]. For simplicity, we confine our consideration to the case of simple shear flow, where the transverse coordinate Y = Xi is measured from the wall (the results are also applicable to turbulent boundary layers). According to Boussinesq s model, the only nonzero component of the Reynolds turbulent shear stress tensor and the divergence of this tensor are defined as... 

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