Notes: The one-point Reynolds stresses have been traditionally expressed in terms of ten tensors. It is, however, known that the independent tensors are only five. We construct a complete set of five orthogonal, traceless, and symmetric second rank tensors in terms of mean strain and vorticity. The system is used to express the one-point Reynolds stresses. The coefficients of the expansion are evaluated in papers VIII and IX. © 1999 American Institute of Physics.

Notes: The basic equations for the two-point Reynolds stresses derived in paper II are solved analytically in two regimes: the UV (ultraviolet) region corresponding to the inertial range and the IR (infrared) region corresponding to k→0. The analytic treatment is possible due to the existence of two smallness parameters: Ui,j(k2νt)−1 in the UV region and kL in the IR region; Ui,j is the mean velocity gradient, νt(k) is the turbulent viscosity, and L is the integral length scale. For an arbitrary flow, the Reynolds stress spectrum in the UV region is given by Eqs. (53545556575859). In the IR region, and in the first-order approximation in kL, the spectra coincide with those of the rapid distortion theory. Since they are flow dependent, we shall discuss a few representative cases. The resulting Reynolds stress spectra, which are shown to reproduce existing data, are the basis for the calculation of the one-point Reynolds stresses to be presented in paper IX. The model has no free parameters. © 1999 American Institute of Physics.

Notes: We present a new expression for the one-point Reynolds stress τij in terms of the strain and vorticity of the large scales. The τij are expressed in terms of only five basic orthogonal tensors rather than the traditional ten tensors. The expression for τij, Eq. (24), contains no adjustable parameters. The derivation of τij is based on the two-point closure dynamic equations for the spectral Reynolds stresses Rij(k) that were developed earlier and the results of which were validated on a wide variety of data comprising shear, buoyancy, two-dimensional (2-D) turbulence, rotation, etc. For the case of homogeneous turbulence, we also derive an expression for the empirical coefficients of the ε equation that depend on the invariants of the flow, the turbulent kinetic energy K and the production P. Examples for special flows are given. The new expressions for τij are shown to reproduce well data from Tavoularis and Corrsin, DNS data, stationary data (pipe flow, channel flow, and homogeneous flow), and the Smagorinsky–Lilly constant, which is shown to be a dynamical variable since it depends on the ratio P/ε and on the invariant {S3}S−3. © 1999 American Institute of Physics.

Notes: The equations determining the linear growth rate ω characterizing a convectively unstable fluid with Rayleigh number Ru bounded below by an impenetrable free boundary and above by a convectively stable fluid with Rayleigh number Rs, are solved numerically. Using the analytical Rayleigh–Bénard growth rate ωRB as a convenient functional form, it is possible to fit the numerical values for ω if the vertical wave number kz = nπ and the Rayleigh number RRB are taken to be functions of Rs, Ru, and the horizontal wave number k⊥, rather than n=integer as in the Rayleigh–Bénard case. In addition, contrary to Rayleigh–Bénard convection, in which the critical Rayleigh number is fixed (RcrRB = 657.5), it is found that Rcru is variable in the presence of a stable layer, i.e., it depends on Rs. Specifically, for Rs 〈 6300, Rcru 〈 RcrRB so that the stable fluid hastens the onset of convection; on the other hand, when Rs (approximately-greater-than) 6300, Rcru (approximately-greater-than) RcrRB so that the system becomes more stable.

Notes: Recently published experimental results [Int. J. Heat Mass Transfer, 23, 738 (1980)] on thermally driven high Rayleigh number turbulent convection have shown that the N∝R2/7 relation (where N is the Nusselt number and R is the Rayleigh number), which is valid up to R≈5×108, is superceded at this point by the relation N=AσR1/3 that holds at least up to R≈1011. For water (Prandtl number σ=6.6), the experimental value for Aσ was found to be Aσ=0.0556±0.001. In the present work, the equations for a turbulent fluid driven by thermal convection are solved using the two-point closure prescription of the direct interaction approximation. The theoretical N vs R relation at high R is found to be of the form N=AσR1/3 and for σ=6.6, and the predicted value of the coefficient Aσ is computed to be Aσ(approximately-less-than)0.08, in good agreement with the experimental value. Extension of the model to situations other than convection is discussed.

Notes: A model for stationary, fully developed turbulence is presented. The physical model used to describe the nonlinear interactions provides an equation for the turbulent spectral energy function F(k) as a function of the time scale for the energy fed into the system, n−1s. The model makes quantitative predictions that are compared with the following available data of a different nature. (a) For turbulent convection, in the case of a constant superadiabatic gradient and for σ(very-much-less-than)1 (σ≡Prandtl number), the convective flux is computed and compared with the result of the mixing length theory (MLT). For the case of a variable superadiabatic gradient, and for arbitrary σ, as in the case of laboratory convection, the Nusselt number N versus Rayleigh number R relation is found to be N=AσR1/3 as recently determined experimentally. The computed Aσ deviates 3% and 8% from recent laboratory data at high R for σ=6.6 and σ=2000. (b) The K–ε and Smagorinsky relations. Four alternative expressions for the turbulent (eddy) viscosity are derived (the K–ε and Smagorinsky relations being two of them) and the numerical coefficients appearing in them are computed. They compare favorably with theoretical estimates (the direct interaction approximation and the renormalization group method), laboratorydata, and simulation studies. (c) The spectral function, transfer term, and dissipation term. The spectral energy function F(k), the transfer term T(k), and the dissipation term νk2F(k) are computed and compared withlaboratory data on grid turbulence. (d) The skewness factor S¯3 is computed and compared with laboratory data. The turbulence model is extended to treat temperature fluctuations characterized by a spectral function G(k). The main results are (e) when both temperature and velocity fluctuations are taken into account, the rate ns(k), that in the first part was taken to be given by the linear mode analysis, can be determined self-consistently from the model itself; (f) in the inertial-convective range, the model predicts the well-known result G(k)∼k−5/3; (g) the Kolmogorov and Batchelor constants are shown to be related by Ba=σt Ko, where σt is the turbulent Prandtl number; and (h) in the inertial-conductive range the model predicts G(k)∼k−17/3 for thermally driven convection as well as for advection of a passive scalar, the difference being contained in the numerical coefficient in front. The predicted G(k) vs k compare favorably with experiments for air (σ=0.725), mercury (σ=0.018), and salt water (σ=9.2).

Notes: We apply a recent model of turbulence to turbulent convection at high Rayleigh number Ra and compare the results with new laboratory and DNS data. We derive a closed set of equations for the total turbulent kinetic energy, turbulent kinetic energy in the z direction, temperature variance, and convective flux. The equations are coupled, time dependent, and nonlocal. We solve the equations both analytically and numerically. In the first case, we neglect diffusion and derive the relation Nu=Nu(σ,Ra), where Nu is the Nusselt number and σ is the Prandtl number. For σ(very-much-greater-than)1, Nu becomes independent of σ; for σ(very-much-less-than)1, Nu is proportional to σ1/3; for 0.025 (mercury)≤σ≤0.7 (helium), Nu is proportional to σ2/7. The numerical solution (with diffusion) yields (a) Nusselt number Nu, 〈θ2〉w, 〈θ2〉c (temperature variance near the wall and at the center), λT (thermal boundary layer thickness), and Pe (Peclet number) versus Ra, (b) the z profile of mean temperature T, 〈θ2〉, horizontal, and vertical Peclet numbers, (c) spectra versus kh (horizontal wave number) of total kinetic energy, vertical kinetic energy, temperature variance, and temperature flux; (d) dependence of the Nu vs Ra relation on the Prandtl number. For large aspect ratios, the agreement with both laboratory and DNS data is satisfactory. The model contains no free parameters. © 1997 American Institute of Physics.

Notes: Using the formalism developed in paper I, we treat the case of shear-driven flows. First, we derive the dynamic equations for the Reynolds stress. The equations are expressed in both tensorial and scalar forms, that is, as a set of coupled differential equations for the functions that enter the expansion of the Reynolds stress in terms of basic tensors. We specialize the general results to (a) axisymmetric contraction, (b) plane strain, and (c) homogeneous shear, for which there is a wealth of DNS, LES, and laboratory data to test the predictions of our model. Second, for homogeneous shear, in the inertial range, the equations for the Reynolds stress spectral function can be solved analytically, E12(k)=−Cε1/3Sk−7/3, which is in excellent agreement with recent data. Since the model has no free parameters, we stress that the model yields a numerical coefficient C, which is also in agreement with the data. Third, we derive the general expressions for the rapid and slow parts of the pressure–strain correlation tensors Πrij and Πsij. Within the second-order closure models, the closure of Πsij (third-order moment) in terms of second-order moments continues to be particularly difficult. The general expression for Πij are then specialized to the three flows discussed above. When Πsij is written in the form first suggested by Rotta, we show that the Rotta constant is a nonconstant tensor. Fourth, we discuss the dissipation tensor εij. In standard turbulence models, one not only assumes that εij=2/3εδij+f(uiuj), where f(x) is a empirical function of the one-point Reynolds stress uiuj, but, in addition, one employs a highly parametrized equation for ε. In the present model, neither of the two assumptions is required nor adjustable parameters are needed since εij is computed directly. The model provides the k-dependent Rij(k) as one of the primary quantities. © 1996 American Institute of Physics.