UNIVERSAL PRANDTL NUMBER IN TWO-DIMENSIONAL KRAICHNAN-BATCHELOR TURBULENCE

2008 ◽  
Vol 22 (20) ◽  
pp. 3421-3431
Author(s):  
MALAY K. NANDY
Keyword(s):  
Prandtl Number ◽  
Mode Coupling ◽  
Perturbation Expansion ◽  
Dynamic Scaling ◽  
Two Dimensional ◽  
Turbulent Prandtl Number ◽  
Energy Regime ◽  
Self Consistent ◽  
Prandtl Numbers

We evaluate the universal turbulent Prandtl numbers in the energy and enstrophy régimes of the Kraichnan-Batchelor spectra of two-dimensional turbulence using a self-consistent mode-coupling formulation coming from a renormalized perturbation expansion coupled with dynamic scaling ideas. The turbulent Prandtl number is found to be exactly unity in the (logarithmic) enstrophy régime, where the theory is infrared marginal. In the energy régime, the theory being finite, we extract singularities coming from both ultraviolet and infrared ends by means of Laurent expansions about these poles. This yields the turbulent Prandtl number σ ≈ 0.9 in the energy régime.

MODE-COUPLING THEORY, DYNAMIC SCALING, AND TWO-DIMENSIONAL TURBULENCE

1995 ◽  
Vol 09 (09) ◽  
pp. 1081-1097 ◽  
Author(s):  
MALAY K. NANDY ◽  
JAYANTA K. BHATTACHARJEE
Keyword(s):  
Perturbation Theory ◽  
Mode Coupling ◽  
Direct Interaction ◽  
External Parameter ◽  
Coupling Scheme ◽  
Dynamic Scaling ◽  
Logarithmic Correction ◽  
Two Dimensional ◽  
Navier Stokes Equation ◽  
Self Consistent

A self-consistent mode-coupling scheme, along with dynamic scaling ideas, is used to obtain a renormalized perturbation theory in the Eulerian framework from Wyld’s perturbation theory of the forced Navier-Stokes equation. For the force-correlation behaving as k−(d−4+y), the Kolmogorov and Kraichnan-Batchelor scaling spectra of two-dimensional turbulence for the inverse energy cascade, [Formula: see text] and the direct entropy cascade, [Formula: see text], are obtained for y=4 and y=6 respectively, including the logarithmic correction for the latter. Unlike the usual Eulerian formulations (e.g. the direct-interaction approximation), the theory is finite in the energy regime, while it becomes marginal in the enstrophy regime, leading to the logarithmic correction. Calculations yield C=6.447 and C′=1.923 at one-loop order, which are in exact agreement with those of field-theoretic renormalization group calculations [P. Olla, Phys. Rev. Lett. 67, 2465 (1991)]. However, a self-consistent treatment of the logarithmic scalings in E(k) and the inverse response-time yields a different value: C′=2.201. The theory is free of any external parameter; the choice of y(=4 or 6) is dictated by the condition of conserved transfer of energy or enstrophy.

Self-Consistent Mode-Coupling Theory, 1/d Expansion and Kolmogorov Turbulence

2003 ◽  
Vol 17 (17) ◽  
pp. 3205-3213 ◽  
Author(s):  
Malay K. Nandy
Keyword(s):  
Mode Coupling ◽  
Perturbation Expansion ◽  
Three Dimensions ◽  
Dynamic Scaling ◽  
Coupling Theory ◽  
Mode Coupling Theory ◽  
Kolmogorov Turbulence ◽  
Self Consistent ◽  
Dimension Expansion ◽  
Kolmogorov Constant

Starting with a self-consistent mode-coupling theory, which was obtained via renormalized perturbation expansion and dynamic scaling ideas, we implement a high d (space dimension) expansion coupled with ∊-expansion. From the theory, which is finite for the Kolmogorov régime, we extract the ultraviolet and infrared poles by means of a double Laurent expansion. The Kolmogorov constant C is found to be C = (40d/27)1/3 in the lowest contributing order of the high d expansion. This yields the value C = 1.64414 in three dimensions.

The heated laminar vertical jet

1967 ◽  
Vol 29 (2) ◽  
pp. 305-315 ◽  
Author(s):  
R. S. Brand ◽  
F. J. Lahey
Keyword(s):  
Boundary Layer ◽  
Temperature Distribution ◽  
Laminar Flow ◽  
Prandtl Number ◽  
Exact Solutions ◽  
Two Dimensional ◽  
Boundary Layer Equations ◽  
Prandtl Numbers ◽  
Vertical Jet ◽  
Steady Laminar

The boundary-layer equations for the steady laminar flow of a vertical jet, including a buoyancy term caused by temperature differences, are solved by similarity methods. Two-dimensional and axisymmetric jets are treated. Exact solutions in closed form are found for certain values of the Prandtl number, and the velocity and temperature distribution for other Prandtl numbers are found by numerical integration.

A Model for Eddy Conductivity and Turbulent Prandtl Number

1973 ◽  
Vol 95 (2) ◽  
pp. 227-234 ◽  
Author(s):  
T. Cebeci
Keyword(s):  
Heat Transfer ◽  
Mass Transfer ◽  
Prandtl Number ◽  
Turbulent Flows ◽  
Pressure Gradients ◽  
Turbulent Prandtl Number ◽  
Temperature Distributions ◽  
Type Flow ◽  
Prandtl Numbers ◽  
Experimental Findings

This paper presents a model for eddy conductivity and turbulent Prandtl number based on the considerations of a Stokes-type flow. The expressions obtained by the model provide continuous velocity and temperature distributions for turbulent flows and are applicable to flows with pressure gradients, mass transfer, and heat transfer. Close to the wall the turbulent Prandtl number appears to be strongly affected by the molecular Prandtl number; away from the wall it is constant, that is, it is independent of the molecular Prandtl number. Calculated results agree well with experiments, including those with fluids having both low and high Prandtl numbers. In addition the results confirm recent experimental findings, in that the mass transfer has no effect on turbulent Prandtl number.

Turbulent Prandtl Number—Where Are We?

1994 ◽  
Vol 116 (2) ◽  
pp. 284-295 ◽  
Author(s):  
William M. Kays
Keyword(s):  
Experimental Data ◽  
Prandtl Number ◽  
Circular Tube ◽  
Two Dimensional ◽  
Turbulent Prandtl Number ◽  
Fully Developed Flow

The objective of this paper is to examine critically the presently available experimental data on Turbulent Prandtl Number for the two-dimensional turbulent boundry layer, and for fully developed flow in a circular tube or a flat duct, and attempt to draw some conclusions as to where matters presently stand.

Thermoconvective Motion of Low Prandtl Number Fluids Within a Horizontal Cylindrical Annulus

1977 ◽  
Vol 99 (4) ◽  
pp. 596-602 ◽  
Author(s):  
J. R. Custer ◽  
E. J. Shaughnessy
Keyword(s):  
Heat Flux ◽  
Natural Convection ◽  
Prandtl Number ◽  
Perturbation Expansion ◽  
Constant Heat Flux ◽  
Basic Flow ◽  
Constant Heat ◽  
Grashof Number ◽  
Cylindrical Annulus ◽  
Prandtl Numbers

Steady natural convection in very low Prandtl number fluids is investigated using a double perturbation expansion in powers of the Grashof and Prandtl numbers. The fluid is contained in a horizontal cylindrical annulus, the walls of which either are held at constant temperature or support a constant heat flux. In both cases the evolution of the flow for increasing Grashof number is of interest. It is found that the basic flow pattern consists of one eddy. For both boundary conditions the center of this eddy falls into the lower half of the annulus as the Grashof number increases. Such behavior is directly opposite to experimental results obtained in fluids of higher Prandtl number.

An Explicit Algebraic Model for Turbulent Buoyant Flows

2003 ◽  
Author(s):  
R. M. C. So ◽  
L. H. Jin ◽  
T. B. Gatski
Keyword(s):  
Heat Flux ◽  
Prandtl Number ◽  
Shear Flows ◽  
Algebraic Model ◽  
Turbulent Prandtl Number ◽  
Flux Transport ◽  
Buoyant Flows ◽  
Model Predictions ◽  
Anisotropy Tensor ◽  
Prandtl Numbers

This paper presents a derivation of an explicit algebraic stress model (EASM) and an explicit algebraic heat flux model (EAHFM) for buoyant shear flows. The models are derived using a projection methodology. The derived EASM has a four-term representation and is applicable to 2-D and 3-D flows. It is an extension of the three-term EASM for incompressible flow and the fourth term is added to account for the effect of buoyancy. The projection methodology is further extended to treat the heat flux transport equation in the derivation of an EAHFM. Again, the weak equilibrium assumption is invoked for the scaled heat flux equation. The basis vector used to represent the scaled heat flux vector is formed with the mean temperature gradient vector and 3×3 tensors, not necessarily symmetric or traceless, deduced from the shear and rotation rate tensors and the stress anisotropy tensor. An explicit algebraic model for buoyant shear flows is then formed with the derived EASM and EAHFM. From the derived EAHFM, an expression for the thermal diffusivity tensor in buoyant shear flows can be deduced. Thus, a turbulent Prandtl number for each of the three heat flux directions can be determined. These Prandtl numbers are functions of the gradient Richardson number. Alternatively, a scalar turbulent Prandtl number can be derived; its value is compared with the directional turbulent Prandtl numbers. The EASM and EAHFM are specialized to calculate 2-D homogeneous buoyant shear flows and the results are compared with direct numerical simulation (DNS) data and other model predictions. Good agreement with DNS data and other model predictions is obtained.

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