Turbulent momentum flux

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 ... 

Turbulent heat transfer is analogous to turbulent momentum transfer. The turbulent momentum flux postulated by Eq. (5-59) carries with it a turbulent... 

We see, as for the species continuity equations, that when we perform the averaging procedure we obtain new dependent variables, in this case the terms pu iu j. These new terms are usually called the turbulent momentum fluxes or the Reynolds stresses, but, as before, we have more dependent variables than equations. Thus, some means to evaluate the turbulent momentum fluxes must be developed. Although no rigorous method of obtaining a closed set of equations is known, a number of semi-empirical approaches have been proposed which yield qualitative or semi-quantitative results for appropriately chosen classes of prr blems. 

The key aspect, then, in numerical simulation of the atmoLj.)heric boundary layer is the evaluation of the turbulent momentum fluxes in the time-averaged equations of motion (24). Considering this, we review briefly some of the more promising techniques that have been used to determine these fluxes. Our objective is not to give a full review, but rather to introduce the types of approaches which in the future may permit the solution of (23) and (24) and thus the prediction of urban wind fields. 

The most common approach to evaluating the turbulent momentum fluxes is to assume that an eddy viscosity Km exists such that... 

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. 

The calculation of the resistance coefficients can be accomplished in the frame of the Monin-Obukhov similarity theory (Monin and Yaglom, 1971). The genuine flux quantities are the friction velocity u and the scale functions 0 and referring to temperature and humidity. The turbulent momentum flux f, the sensible heat flux the mass flux from evaporation and condensation and the corresponding latent heat flux are... 

The final form of the Reynolds stress transport equation (i.e., the transport equation for the turbulent momentum flux ) is ... 

The physical interpretation of the terms in the equation is not necessary obvious. The first term on the LHS denotes the rate of accumulation of the kinematic turbulent momentum flux within the control volume. The second term on the LHS denotes the advection of the kinematic turbulent momentum flux by the mean velocity. In other words, the left hand side of the equation constitutes the substantial time derivative of the Re3molds stress tensor The first and second terms on the RHS denote the production... 

Launder and Spalding [94] argued that the pressure diffusion terms and the molecular diffusion of turbulent momentum fluxes are smaller than the rest of... 

The terms in this equation have physical interpretations analogous to those in the momentum flux equation (1.394), except for the additional term (i.e., the second term on the RHS), which is a production/loss term related to the mean scalar quantity gradient. Physically, this term suggests production of the scalar quantity flux when there is a momentum flux in a mean scalar quantity gradient. The turbulent momentum flux implies a turbulent movement of the fluid. If that movement occurs across a mean scalar quantity gradient, then the scalar quantity fluctuation would be expected. 

These are the components of the turbulent momentum flux and are called Reynolds stresses. 

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. 

The physical interpretation of the terms in the equation is not necessary obvious. The first term on the LHS denotes the rate of accumulation of the kinematic turbulent momentum flux within the control volume. The second term on the LHS denotes the advection of the kinematic turbulent momentum flux by the mean velocity. In other words, the left hand side of the equation constitutes the substantial time derivative of the Reynolds stress tensor v v. The first and second terms on the RHS denote the production of the kinematic turbulent momentum flux by the mean velocity shears. The third term on the RHS denotes the transport of the kinematic momentum flux by turbulent motions (turbulent diffusion). This latter term is unknown and constitutes the well known moment closure problem in turbulence modeling. The fourth and fifth terms on the RHS denote the turbulent transport by the velocity-pressure-gradient correlation terms (pressure diffusion). The sixth term on the RHS denotes the redistribution by the return to isotropy term. In the engineering literature this term is called the pressure-strain correlation, but is nevertheless characterized by its redistributive nature (e.g., [132]). The seventh term on the RHS denotes the molecular diffusion of the turbulent momentum flux. The eighth term on the RHS denotes the viscous dissipation term. This term is often abbreviated by the symbol... 

For practical applications, second-order closure models are required for the third-order diffusion correlations, the pressure-strain correlation and the dissipation rate correlation as described by Launder and Spalding [95] and Wilcox ([185], Sect. 6.3). Launder and Spalding [95] argued that the pressure diffusion terms and the molecular diffusion of turbulent momentum fluxes are smaller than the rest of the terms in the equation. These terms can thus be sufficiently approximated by a gradient... 

As velocity continues to rise, the thicknesses of the laminar sublayer and buffer layers decrease, almost in inverse proportion to the velocity. The shear stress becomes almost proportional to the momentum flux (pk ) and is only a modest function of fluid viscosity. Heat and mass transfer (qv) to the wall, which formerly were limited by diffusion throughout the pipe, now are limited mostly by the thin layers at the wall. Both the heat- and mass-transfer rates are increased by the onset of turbulence and continue to rise almost in proportion to the velocity. 

In a free jet the absence of a pressure gradient makes the momentum flux at any cross section equal to the momentum flux at the inlet, ie, equations 16 and 17 define jet velocity at all points. For a cylindrical jet this leads to a center-line velocity that varies inversely with (x — aig), whereas for slot jets it varies inversely with the square root of (x — Xq As the jet proceeds still further downstream the turbulent entrainment initiated by the jet is gradually subordinated to the turbulence level in the surrounding stream and the jet, as such, disappears. 

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. 

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... 

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. 

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... 

The expression —gwxw y (SI units N/m2) is an averaged momentum flow per unit area, and so comparable to a shear stress A force in the direction of the y-axis acts at a surface perpendicular to the a -axis. Terms of the general form —gwf-wij are called Reynolds stresses or turbulent stresses. They are symmetrical tensors. In a corresponding manner, the energy equation (3.135), contains a turbulent heat flux of the form... 

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... 

The turbulence in the surface mixed layer is associated with a vertical momentum flux, the Reynolds stress, which has a vertical gradient. An equilibrium can develop in the surface mixed layer when the vertical gradient of the Reynolds stress and the Coriolis force acting on the water parcels moving in the surface mixed layer balance each other. 

An alternative procedure is to simplify the transport equation for the kinematic momentum fluxes or Re molds stresses. The contraction of the Re3molds stress transport equation (1.394) (that is, when the 3 equations for the 3 normal stresses, i = k = 1,2,3) are summed up) gives an exact transport equation for the turbulent kinetic energy (e.g., [167] [131] [106]). 

The general transport equation for the specific turbulent fluxes of scalar variables is derived in analogy to the corresponding momentum flux equations, i.e., the Reynolds stress equations. The derivation combines two equations for the fluctuations to produce a flux equation. For the first equation we start with the momentum fluctuation equation (1.389), multiply it by the scalar quantity perturbation ip, and Reynolds average ... 

To proceed we need to put up dimensionless relations for the heat and mass transfer fluxes in the turbulent boundary layer using a procedure analoguous to the one applied for the momentum flux (5.249) in which the Boussinesq s turbulent viscosity hypothesis is involved. 

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