Coefficient turbulent viscosity

Chemical potential, dynamic viscosity, mean Stoichiometric coefficient Turbulent viscosity... 

Usually, however, the stresses are modeled with the help of a single turbulent viscosity coefficient that presumes isotropic turbulent transport. In the RANS-approach, a turbulent or eddy viscosity coefficient, vt, covers the momentum transport by the full spectrum of turbulent scales (eddies). Frisch (1995) recollects that as early as 1870 Boussinesq stressed turbulence greatly increases viscosity and proposed an expression for the eddy viscosity. The eventual set of equations runs as... 

Note that the Eqs. (1), (2), and (8) are really and essentially different due to the absence or presence of different turbulent transport terms. Only by incorporating dedicated formulations for the SGS eddy viscosity can one attain that LES yield the same flow field as DNS. RANS-based simulations with their turbulent viscosity coefficient, however, essentially deliver steady flow fields and as such are never capable of delivering the same velocity fields as the inherently transient LES or DNS, irrespectively of the refinement of the computational grid ... 

Prandtl s mixing length hypothesis (Prandtl, 1925) was developed for momentum transport, instead of mass transport. The end result was a turbulent viscosity, instead of a turbulent diffusivity. However, because both turbulent viscosity and turbulent diffusion coefficient are properties of the flow field, they are related. Turbulent viscosity describes the transport of momentum by turbulence, and turbulent diffusivity describes the transport of mass by the same turbulence. Thus, turbulent viscosity is often related to turbulent diffusivity as... 

The equation of motion and the equation of energy balance can also be time averaged according to the procedure indicated above (SI, pp. 336 et seq. G7, pp. 191 et seq. pp. 646 et seq.). In this averaging process there arises in the equation of motion an additional component to the stress tensor t(,) which may be written formally in terms of a turbulent (eddy) coefficient of viscosity m(I) and in the equation of energy balance there appears an additional contribution to the energy flux q(1), which may be written formally in terms of the turbulent (eddy) coefficient of thermal conductivity Hence for an incompressible fluid, the x components of the fluxes may be written... 

The difference between this equation for turbulent flow and the Navier-Stokes equation for laminar flow is the Reynolds stress/turbulent stress term —pujuj appears in the equation of motion for turbulent flow. This equation of motion for turbulent flow involves non-linear terms, and it is impossible to be solved analytically. In order to solve the equation in the same way as the Navier-Stokes equation, the Reynolds stress or fluctuating velocity must be known or calculated. Two methods have been adopted to avoid this problem—phenomenological method and statistical method. In the phenomenological method, the Reynolds stress is considered to be proportional to the average velocity gradient and the proportional coefficient is considered to be turbulent viscosity or mixing length ... 

When the transport is considered without turbulence we have, in general, Dj- u is the cinematic viscosity for the momentum transport a = A,/(pCp) is the thermal diffusivity and D is the diffusion coefficient of species A. Whereas with turbulence we have, in general, Dj-, w, is the cinematic turbulence viscosity for the momentum transport a, =, /(pCp) is the thermal turbulence diffusivity and D t is the coefficient of turbulent diffusion of species A frequently = a = D t due to the hydrodynamic origin of the turbulence. 

The formulation of Section 9.5.1 has served to remove the chemistry from the field equations, replacing it by suitable jump conditions across the reaction sheet. The expansion for small S/l, subsequently serves to separate the problem further into near-field and far-field problems. The domains of the near-field problems extend over a characteristic distance of order S on each side of the reaction sheet. The domains of the far-field problems extend upstream and downstream from those of the near-field problems over characteristic distances of orders from to /. Thus the near-field problems pertain to the entire wrinkled flame, and the far-field problems pertain to the regions of hydrodynamic adjustment on each side of the flame in essentially constant-density turbulent flow. Either matched asymptotic expansions or multiple-scale techniques are employed to connect the near-field and far-field problems. The near-field analysis has been completed for a one-reactant system with allowance made for a constant Lewis number differing from unity (by an amount of order l/P) for ideal gases with constant specific heats and constant thermal conductivities and coefficients of viscosity [122], [124], [125] the results have been extended to ideal gases with constant specific heats and constant Lewis and Prandtl numbers but thermal conductivities that vary with temperature [126]. The far-field analysis has been... 

Just like the turbulent viscosity, the turbulent fractions of the thermal dif-fusivity, thermal conductivity and the diffusion coefficient have to disappear at the solid wall. In contrast, at some distance away from the wall the turbulent exchange is far more intensive than the molecular motion. This leads to good mixing of the fluid particles. The result of this is that the velocity, temperature and concentration profiles are more uniform in the core than those in laminar flows, as shown in Fig. 3.15 for the velocity profiles in flow over a body. 

The coefficient of turbulent viscosity vr is the most important among them. A number of scientific efforts were spent for estimating it with respect to the whole wide spectrum of problems of fluid mechanics and heat and mass exchange. Thousands of lengthy papers have been devoted to the problem of turbulence . However, we still have no any sufficiently grounded and generally accepted theory. Turbulence still remains to be the most serious challenge to theoretical physics. 

Note that the turbulent viscosity parameter has an empirical origin. In connection with a qualitative analysis of pressure drop measurements Boussinesq [19] introduced some apparent internal friction forces, which were assumed to be proportional to the strain rate ([20], p 8), to fit the data. To explain these observations Boussinesq proceeded to derive the same basic equations of motion as had others before him, but he specifically considered the molecular viscosity coefficient to be a function of the state of flow and not only on the system properties [135]. It follows that the turbulent viscosity concept (frequently referred to as the Boussinesq hypothesis in the CFD literature) represents an empirical first attempt to account for turbulence effects by increasing the viscosity coefficient in an empirical manner fitting experimental data. Moreover, at the time Boussinesq [19] [20] was apparently not aware of the Reynolds averaging procedure that was published 18 years after the first report by Boussinesq [19] on the apparent viscosity parameter. 

In general, In fully developed turbulent flows the turbulent viscosity, conductivity and diffusivity vary from point to point in the flow field and are many magnitudes greater than the molecular coefficients. 

However, to solve the heat and mass transfer equations an additional modeling problem has to be overcome. While there are sufficient measurements of the turbulent velocity field available to validate the different i>t modeling concepts proposed in the literature, experimental difficulties have prevented the development of any direct modeling concepts for determining the turbulent conductivity at, and the turbulent diffusivity Dt parameters. Nevertheless, alternative semi-empirical modeling approaches emerged based on the hypothesis that it might be possible to calculate the turbulent conductivity and diffusivity coefficients from the turbulent viscosity provided that sufficient parameterizations were derived for Prj and Scj. 

Formulas (1.6.1) and (1.6.2) have formed the basis of most theoretical investigations on the determination of the average fluid velocity and the drag coefficient in the stabilized region of turbulent flow in a circular tube (and a plane channel of width 2a). The corresponding results obtained on the basis of Prandtl s relation (1.1.21) and von Karman s relation (1.1.22) for the turbulent viscosity can be found in [276,427]. In what follows, major attention will be paid to empirical and semiempirical formulas that approximate numerous experimental data quite well. 

Turbulent Prandtl number. In a number of problems, it is important to know, as a rule, only one component of turbulent transfer, namely, the component normal to the wall. In this case, the strength of turbulent diffusion is characterized by a single scalar quantity Dt, just as the intensity of turbulent transfer of momentum is characterized by a single scalar quantity vt. The ratio of the turbulent viscosity coefficient to the turbulent diffusion coefficient,... 

The simplest way to close equations (3.1.37) is to use the hypothesis that the turbulent Prandtl number for the examined process is a constant quantity. Then it readily follows from Eq. (3.1.39) that the turbulent diffusion coefficient is proportional to the turbulent viscosity >t = i /Pr,. By using the expression for vx borrowed from the corresponding hydrodynamic model, one can obtain the desired value of Dt. In particular, following Prandtl s or von Karman s model, one can use formula (1.1.21) or (1.1.22) for vx. 

If V becomes too large, then the ratio of inertial to viscous forces (characterized by a Reynolds number. Re = pva/fi, where // is a representative average value of the coefficient of viscosity) becomes large enough to cause the flow to be turbulent. Turbulence occurs in most practical burners and introduces qualitative differences from the present predictions the flame fluctuates rapidly and on the average is thicker, with hja approximately independent of i or a [12]. This flame-height behavior may be obtained from the present analysis by replacing D by a turbulent diffusivity that is assumed to be proportional to the product va. However, there are many properties of turbulent diffusion flames that cannot be predicted well by a simple approach of this kind (see Section 10.2). 

Although Ey and are analogous to fj. and v, respectively, in that all these quantities are coefficients relating shear stress and velocity gradient, there is a basic difference between the two kinds of quantities. The viscosities n and v are true properties of the fluid and are the macroscopic result of averaging motions and momenta of myriads of molecules. The eddy viscosity and the eddy diffusivity are not just properties of the fluid but depend on the fluid velocity and the geometry of the system. They are functions of all factors that influence the detailed patterns of turbulence and the deviating velocities, and they are especially sensitive to location in the turbulent field and the local values of the scale and intensity of the turbulence. Viscosities can be measured on isolated samples of fluid and presented in tables or charts of physical properties, as in Appendixes 8 and 9. Eddy viscosities and diffusivities are determined (with difficulty, and only by means of special instruments) by experiments on the flow itself. 

The SI unit for the coefficient of viscosity ps kg m s or N s m the commonly used unit, the poise, is one-tenth of the SI unit. The type of flow to which (11.95) applies is called laminar, streamline, or Newtonian flow. In flow of this kind there is a net component of velocity in the direction of flow superimposed on the random molecular velocities. Streamline flow is observed if the velocity of flow is not too large with very rapid flow the motion becomes turbulent and (11.95) no longer applies. 

In the formula xeff = x + xt xt- Turbulent viscosity coefficient x- Coefficient of kinematic viscosity n- Porosity of porous medium. 

If one supposes that turbulence diffusion coefficient Dt is equal to kinematic coefficient of turbulence viscosity Vt which in its turn can be expressed by specific kinetic turbulence energy K (mVsec ) and rate of dissipation of last one is e (mVsec ) then (1.16) will be as following ... 

Since intensive mixing of liquid flows is observed at developed turbulent regime, so according to [136-139] one can t directly apply ordinary equations of continuum movement (for example, Navie-Stocks equation with molecular coefficient of viscosity) in this case. That is why it is generally accepted to use the so-called average equations of turbulent continuum movement, they are also called as average by Remolds. 

The form of Eq. (5.9) models a retum-to-isotropy effect due to fluctuating interfacial momentum coupling and reduces the turbulent viscosity from that predicted by the single-phase model. The turbulence energy exchange rate coefficient Ey is given by... 

The theoretical description of the turbulent mixing of reactants in tubular devices is based on the following model assumptions the medium is a Newtonian incompressible medium, and the flow is axis-symmetrical and nontwisted turbulent flow can be described by the standard model [16], with such parameters as specific kinetic energy of turbulence K and the velocity of its dissipation e and the coefficient of turbulent diffusion is equal to the kinematic coefficient of turbulent viscosity D, = Vj- =... 

If the turbulent diffusion coefficient is assumed to be equal to the kinematic coefficient of turbulent viscosity Vj-, which in turn, can be expressed via the specific kinetic energy of turbulence K and its dissipation rate e, then Equation 2.13 will... 

Temperature conductivity coefficient Dynamic coefficient of turbulent viscosity Kinematic coefficient of turbulent viscosity Characteristic time of turbulent mixing A pressure drop Input-output pressure drop... 

Under the conditions prescribed, the methods outlined above always give identical values for the coefficients of viscosity of pure liquids assuming that turbulence does not exist and that Newton s law connecting velocity gradient and shear stress truly represents the conditions. This is actually the case, as is known from experience, for the majority of pure liquids, but there are exceptions, which appear important for our purpose because they concern liquids made up of long chain substances as we shall see later, these play an important part in the realm of high polymer solutions. It has been proved impossible in many cases to describe the me-... 

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