Turbulent flow momentum flux

In turbulent flow, momentum transfer is often characterized by an eddy viscosity Ey defined by analogy with Newton s Law of Viscosity. The time averaged momentum flux (shear stress) in the y direction owing to the gradient of the time averaged velocity in Ak z direction is given by... 

This expression applies to the transport of any conserved quantity Q, e.g., mass, energy, momentum, or charge. The rate of transport of Q per unit area normal to the direction of transport is called the flux of Q. This transport equation can be applied on a microscopic or molecular scale to a stationary medium or a fluid in laminar flow, in which the mechanism for the transport of Q is the intermolecular forces of attraction between molecules or groups of molecules. It also applies to fluids in turbulent flow, on a turbulent convective scale, in which the mechanism for transport is the result of the motion of turbulent eddies in the fluid that move in three directions and carry Q with them. 

The situation with regard to convective (turbulent) momentum transport is somewhat more complex because of the tensor (dyadic) character of momentum flux. As we have seen, Newton s second law provides a correspondence between a force in the x direction, Fx, and the rate of transport of x-momentum. For continuous steady flow in the x direction at a bulk... 

The foregoing procedure can be used to solve a variety of steady, fully developed laminar flow problems, such as flow in a tube or in a slit between parallel walls, for Newtonian or non-Newtonian fluids. However, if the flow is turbulent, the turbulent eddies transport momentum in three dimensions within the flow field, which contributes additional momentum flux components to the shear stress terms in the momentum equation. The resulting equations cannot be solved exactly for such flows, and methods for treating turbulent flows will be considered in Chapter 6. 

These turbulent momentum flux components are also called Reynolds stresses. Thus, the total stress in a Newtonian fluid in turbulent flow is composed of both viscous and turbulent (Reynolds) stresses ... 

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... 

Authors efforts in this part of the work have been concentrated on developing turbulence closures for the statistical description of two-phase turbulent flows. The primary emphasis is on development of models which are more rigorous, but can be more easily employed. The main subjects of the modeling are the Reynolds stresses (in both phases), the cross-correlation between the velocities of the two phases, and the turbulent fluxes of the void fraction. Transport of an incompressible fluid (the carrier gas) laden with monosize particles (the dispersed phase) is considered. The Stokes drag relation is used for phase interactions and there is no mass transfer between the two phases. The particle-particle interactions are neglected the dispersed phase viscosity and pressure do not appear in the particle momentum equation. 

The present approach to the prediction of thermal transport in turbulent flow neglects the effect of thermal flux and temperature distribution upon the relationship of thermal to momentum transport. The influence of the temperature variation upon the important molecular properties of the fluid in both momentum and thermal transport may be taken into account without difficulty if such refinement is necessary. 

In Table IV, we see that established techniques for velocity measurement allow us to determine the average momentum flux, average velocity, turbulent intensities, and shear stress. Next on the list, to complete the flow field description, is the fluctuation mass flux, and first on the combustion field list is the temperature and major species densities of the flame gases. 

The value of Km depends on the properties of the mean flow at a particular location and time. To account for the contribution of thermal stratification (buoyancy) to the production or suppression of turbulent energy. Km is taken to be a function of the local value of the flux Richardson number, which expresses the ratio of the rate of generation of energy by buoyancy forces to the rate of generation of energy by the turbulent momentum fluxes. In this approach the influence of the past history of the turbulence on velocity field is not considered the approach is termed a local theory. 

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... 

Analogies Between Mass, Heat and Momentum Transfer Fluxes A comparison of the partial differential equations for the conservation of heat, mass and momentum in a turbulent flow field (5.240), (5.241) and (5.242) shows that the equations are mathematically similar provided that the pressure term in the momentum equation is negligible [135]. If the corresponding boundary contitions are similar too, the normalized solution of these equations will have the same form. 

This represents the ratio of the momentum flux carried by the fluid axially along the tube to the viscous momentum flux transported normal to the flow in the radial direction. Laminar flows are dominated by the fluid viscosity and are stable, whereas turbulent flows are dominated by the fluid density inertia and are unstable. 

On pages 92 to 98 tliie distribution of velocity and its accompanying momentum flux in a flowing stream in turbulent flow through a pipe was described. Three rather ill-defined zones in the cross section of the pipe were identified. In the first, immediately next to the wall, eddies are rare, and momentum flow occurs almost entirely by viscosity in the second, a mixed regime of combined viscous and turbulent momentum transfer occurs in the main part of the stream, which occupies the bulk of the cross section of the stream, only the momentum flow generated by the Reynolds stresses of turbulent flow is important. The three zones are called the viscous sublayer, the buffer zone, and the turbulent core, respectively. 

The situation is analogous to momentum flux, where the relative Importance of turbulent shear to viscous shear follows the same general pattern. Under certain ideal conditions, the correspondence between heat flow and momentum flow is exact, and at any specific value of rjr the ratio of heat transfer by conduction to that by turbulence equals the ratio of momentum flux by viscous forces to that by Reynolds stresses. In the general case, however, the correspondence is only approximate and may be greatly in error. The study of the relationship between heat and momentum flux for the entire spectrum of fluids leads to the so-called analogy theory, and the equations so derived are called analogy equations. A detailed treatment of the theory is beyond the scope of this book, but some of the more elementary relationships are considered. 

Turbulent heat transfer is similar to turbulent momentum transfer. The heat flux in turbulent flow is thus comprised of molecular and turbulent components. 

However, there, are additional terms of the form p which are not zero. These terms, components of the turbulent momentum flux, are called Reynolds stresses. Their nature is such that they must be handled semiempirically. As such, therefore, the approach used for frictional heating in laminar flow cannot be used. 

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