Shear stress total, turbulent

In general, the time-averaged value of the product of the fluctuations is non-zero so there is an additional flux of x-momentum in the y-direction due to the velocity fluctuations v x and v y. This momentum flux is equivalent to an extra apparent shear stress acting in the x-direction on the plane normal to the y-coordinate direction. Consequently, the mean total shear stress for turbulent flow can be written as... 

Equation (5-108) expresses the total heat conduction as a sum of the molecular conduction and the macroscopic eddy conduction. In a similar fashion, the shear stress in turbulent flow could be written... 

Eddy momentum diffusivity in turbulent flow. In Section 3.10F the total shear stress for turbulent flow was written as follows when the molecular and turbulent contributions are summed together ... 

From equation 1.41, the total shear stress varies linearly from a maximum fw at the wall to zero at the centre of the pipe. As the wall is approached, the turbulent component of the shear stress tends to zero, that is the whole of the shear stress is due to the viscous component at the wall. The turbulent contribution increases rapidly with distance from the wall and is the dominant component at all locations except in the wall region. Both components of the mean shear stress necessarily decline to zero at the centre-line. (The mean velocity gradient is zero at the centre so the mean viscous shear stress must be zero, but in addition the velocity fluctuations are uncorrelated so the turbulent component must be zero.)... 

Viscous and turbulent contributions to the total shear stress for flow in a pipe... 

Equations 2.3 to 2.6 are true, irrespective of the nature of the fluid. They are also valid for both laminar and turbulent flow. In the latter case, the shear stress is the total shear stress comprising the viscous stress and the Reynolds stress. 

In outlining the main steps in obtaining an analogy solution, attention will here be given to two-dimensional flow. The total shear stress and total heat transfer rate are made up of the molecular and turbulent contributions, i.e. ... 

For y > 5 the turbulence stress and heat transfer rate become important However, near the wall the total shear stress and total heat transfer rate will remain effectively constant and equal to the wall shear stress and wall heat transfer rate, respectively. 

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 ... 

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... 

Nash (Nl) has used a combination of Bradshaw s MTES ideas and a Newtonian assumption to treat three-dimensional turbulent boundary layers. Nash takes Bradshaw s structural assumption for the total shear stress vector. 

In which et is the turbulent viscosity or eddy diffusivity for momentum transfer (SI units m2/s). This then allows the total shear stress to be expressed by... 

During food engineering operations, many fluids deviate from laminar flow when subjected to high shear rates. The resulting turbulent flow gives rise to an apparent increase in viscosity as the shear rate increases in laminar flow, i.e., shear stress = viscosity x shear rate. In turbulent flow, it would appear that total shear stress = (laminar stress + turbulent stress) x shear rate. The most important part of turbulent stress is related to the eddies diffusivity of momentum. This can be recognized as the atomic-scale mechanism of energy conversion and its redistribution to the dynamics of mass transport processes, responsible for the spatial and temporal evolution of the food system. 

Subsequent experimental validation lead to additional partitioning of the inner constant stress layer according to the relative magnitude of the viscous and turbulent components of the total shear stress. The total shear stress takes the form ... 

The total shear stress in a turbulent fluid is the sum of the viscous stress and the turbulent stress, or... 

The existence of turbulent flow can be advantageous in the sense of providing increased heat and mass transfer rates. However, the motion is extremely complicated and difficult to describe theoretically [3, 8]. In dealing with turbulent flow it is customary to speak of a total shear stress and total fluxes normal to the main flow direction (the main flow is in the x direction, and the y axis is normal to the flow direction), which are defined as... 

Fluid flow past a surface or boundary leads to surface forces acting on it. These surface forces depend on the rate at which fluid is strained by the velocity field. A stress tensor with nine components is used to describe the surface forces on a fluid element. The tangential component of the surface forces with respect to the boundary is known as shear stress. The nature or origin of shear stress depends on the nature of flow, i. e., laminar or turbulent. The stress components for a laminar flow are functions of the viscosity of the fluid and are known as viscous stresses. The turbulent flow has additional contributions known as Reynolds stresses due to velocity fluctuation, i. e., the stresses of a laminar flow are increased by additional stresses known as apparent or Reynolds stresses. Hence, the total shear stresses for a turbulent flow are the sum of viscous stresses and apparent stresses. In a turbulent flow, the apparent stresses may outweigh the viscous conponents. 

The turbulent energy, which is produced in a turbulent regime, is related to the total shear stress, but the drag force is then not effective in maintaining the particles in sus pension. 

Particle and aggregate settling speeds are typically larger than the actual deposition rates of these particles onto bottom sediments. The total particle deposition flux to the bed J (g/m /s) (see Equation 10.7) is equivalent to the particulate matter settling flux modified by a probability of deposition that is almost always less than one. Turbulence at the interface may act as a barrier to the attachment of settling material. As shown in Figures 10.3 and 10.4, the probability of deposition is a function of shear stress. There is a range of bottom shear stresses for which the probability of deposition is less than one those particles that do not deposit will remain entrained or suspended in the flow. 

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