Turbulence definition

Noncircular Channels Calciilation of fric tional pressure drop in noncircular channels depends on whether the flow is laminar or tumu-lent, and on whether the channel is full or open. For turbulent flow in ducts running full, the hydraulic diameter shoiild be substituted for D in the friction factor and Reynolds number definitions, Eqs. (6-32) and (6-33). The hydraiilic diameter is defined as four times the channel cross-sectional area divided by the wetted perimeter. For example, the hydraiilic diameter for a circiilar pipe is = D, for an annulus of inner diameter d and outer diameter D, = D — d, for a rectangiilar duct of sides 7, h, Dij = ah/[2(a + h)].T ie hydraulic radius Rii is defined as one-fourth of the hydraiilic diameter. 

Re using the equivalent diameters defined in the following. This situation is, by arbitrary definition, opposite to that for the hydraulic diameter used for turbulent flow. 

The value of tire heat transfer coefficient of die gas is dependent on die rate of flow of the gas, and on whether the gas is in streamline or turbulent flow. This factor depends on the flow rate of tire gas and on physical properties of the gas, namely the density and viscosity. In the application of models of chemical reactors in which gas-solid reactions are caiTied out, it is useful to define a dimensionless number criterion which can be used to determine the state of flow of the gas no matter what the physical dimensions of the reactor and its solid content. Such a criterion which is used is the Reynolds number of the gas. For example, the characteristic length in tire definition of this number when a gas is flowing along a mbe is the diameter of the tube. The value of the Reynolds number when the gas is in streamline, or linear flow, is less than about 2000, and above this number the gas is in mrbulent flow. For the flow... 

Fundamental, laminar, and turbulent burning velocities describe three modes of flame propagation (see the Glossary for definitions). The fundamental burning velocity, S, is as its name implies, a fundamental property of a flammable mixture, and is a measure of how fast reactants are consumed and transformed into products of combustion. Fundamental burning velocity data for selected gases and vapors are listed in Appendix C of NFPA68 (1998). 

Here C = aa ) is the covariance of the basis functions used to model the turbulence. Covariance matrices are positive semi-definite by dehnition which implies a C a > 0, and thus a dehned maximum of Pr a exists. 

If the turbulent flame is ever proven to have asymptotically a constant flame brush thickness and constant speed in constant, i.e., nondecaying, turbulence, then the aforementioned turbulent flame speed and the flame brush thickness (5 give a well-defined sufficient characterization of the flame in its asymptotic behavior. However, it is not proven up to now that the studied experimental devices have been large enough to ensure that this asymptotic state can be reached. Besides, the correct definitions for the turbulent flame speed or flame brush thickness, as given above, are far from... 

A very fine space resolution is required to measure the gradient of turbulent velocity fluctuations and calculate turbulent dissipation directly from the definition [5, 6]. 

The lack of hydrodynamic definition was recognized by Eucken (E7), who considered convective diffusion transverse to a parallel flow, and obtained an expression analogous to the Leveque equation of heat transfer (L5b, B4c, p. 404). Experiments with Couette flow between a rotating inner cylinder and a stationary outer cylinder did not confirm his predictions (see also Section VI,D). At very low rotation rates laminar flow is stable, and does not contribute to the diffusion process since there is no velocity component in the radial direction. At higher rotation rates, secondary flow patterns form (Taylor vortices), and finally the flow becomes turbulent. Neither of the two flow regimes satisfies the conditions of the Leveque equation. 

Its first invariant A] is equal to zero by definition. The second and third invariants of this tensor arc A — atJaJt and A3 — atjajkakt, respectively. The range of physically allowed values of A2 and A3 is bounded and represented by the so-called Lumley triangle in the (A3, A2) plane (Lumley, 1978). The distanced = (Ay + Af) from the isotropic state, i.e., from the origin (A2 — 0, A3 — 0), is a measure of the degree of anisotropy. See also Escudie and Line (2006) for a more extensive discussion as to how to quantify and visualize how different from isotropic turbulence a stirred vessel is. 

In this definition, ps and pt are the solid and fluid densities, respectively. The characteristic diameter of the particles is ds (which is used in calculating the projected cross-sectional area of particle in the direction of the flow in the drag law). The kinematic viscosity of the fluid is vf and y is a characteristic strain rate for the flow. In a turbulent flow, y can be approximated by l/r when ds is smaller than the Kolmogorov length scale r. (Unless the turbulence is extremely intense, this will usually be the case for fine particles.) Based on the Stokes... 

Using the definition for the turbulent viscosity (jit — /An /xmoi), which gives a result similar to the standard k-s model with only a small difference in the modeling constant, the effective viscosity is now defined as a function of k and s in Eq. (16) in algebraic form. 

The value of the Reynolds number which approximately separates laminar from turbulent flow depends, as previously mentioned, on the particular configuration of the system. Thus the critical value is around 50 for a film of liquid or gas flowing down a flat plate, around 500 for flow around a sphere, and around 2500 for flow through a pipe. The characteristic length in the definition of the Reynolds number is, for example, the diameter of the sphere or of the pipe in two of these examples. 

Equation (e) is merely a definition of the mass flow rate. Equation (/) is a standard correlation for the friction factor for turbulent flow. (Note that the correlation between /and the Reynold s number (Re) is also available as a graph, but use of data from a graph requires trial-and-error calculations and rules out an analytical solution.)... 

By definition, the turbulent kinetic energy k can be found directly from the turbulent energy spectrum by integrating over wavenumber space ... 

From the definition of the turbulent energy spectrum, e is related to Eu k, t) by... 

From this definition, it can be observed that T,(k. t) is the net rate at which turbulent kinetic energy is transferred from wavenumbers less than k to wavenumbers greater than k. In fully developed turbulent flow, the net flux of turbulent kinetic energy is from large to small scales. Thus, the stationary spectral energy transfer rate Tu(k) will be positive at spectral equilibrium. Moreover, by definition of the inertial range, the net rate of transfer through wavenumbers /cei and kdi will be identical in a fully developed turbulent flow, and thus... 

By definition of the dissipation range, s even when the turbulence is not in spectral equilibrium. 

Note that hv operates on the random field U(r, f) and (for fixed parameters V, x, and t) produces a real number. Thus, unlike the LES velocity PDF described above, the FDF is in fact a random variable (i.e., its value is different for each realization of the random field) defined on the ensemble of all realizations of the turbulent flow. In contrast, the LES velocity PDF is a true conditional PDF defined on the sub-ensemble of all realizations of the turbulent flow that have the same filtered velocity field. Hence, the filtering function enters into the definition of /u u(V U ) only through the specification of the members of the sub-ensemble. 

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