scholarly journals Turbulent Prandtl number and characteristic length scales in stably stratified flows: steady-state analytical solutions

2021 ◽  
Author(s):  
Sukanta Basu ◽  
Albert A. M. Holtslag
Keyword(s):  
Prandtl Number ◽  
Characteristic Length ◽  
Stratified Flows ◽  
Turbulent Prandtl Number ◽  
Length Scales ◽  
Functional Relationships ◽  
Stably Stratified Flows ◽  
Theoretical Predictions ◽  
The Stability ◽  
Theoretical Foundations

AbstractIn this study, the stability dependence of turbulent Prandtl number ($$Pr_t$$ P r t ) is quantified via a novel and simple analytical approach. Based on the variance and flux budget equations, a hybrid length scale formulation is first proposed and its functional relationships to well-known length scales are established. Next, the ratios of these length scales are utilized to derive an explicit relationship between $$Pr_t$$ P r t and gradient Richardson number. In addition, theoretical predictions are made for several key turbulence variables (e.g., dissipation rates, normalized fluxes). The results from our proposed approach are compared against other competing formulations as well as published datasets. Overall, the agreement between the different approaches is rather good despite their different theoretical foundations and assumptions.

scholarly journals On the turbulent Prandtl number in homogeneous stably stratified turbulence

2010 ◽  
Vol 644 ◽  
pp. 359-369 ◽  
Author(s):  
SUBHAS K. VENAYAGAMOORTHY ◽  
DEREK D. STRETCH
Keyword(s):  
Kinetic Energy ◽  
Prandtl Number ◽  
Turbulent Kinetic Energy ◽  
Decay Time ◽  
Time Scale ◽  
Richardson Number ◽  
Stratified Flows ◽  
Length Scale ◽  
Stratified Turbulence ◽  
Turbulent Prandtl Number

In this paper, we derive a general relationship for the turbulent Prandtl number Prt for homogeneous stably stratified turbulence from the turbulent kinetic energy and scalar variance equations. A formulation for the turbulent Prandtl number, Prt, is developed in terms of a mixing length scale LM and an overturning length scale LE, the ratio of the mechanical (turbulent kinetic energy) decay time scale TL to scalar decay time scale Tρ and the gradient Richardson number Ri. We show that our formulation for Prt is appropriate even for non-stationary (developing) stratified flows, since it does not include the reversible contributions in both the turbulent kinetic energy production and buoyancy fluxes that drive the time variations in the flow. Our analysis of direct numerical simulation (DNS) data of homogeneous sheared turbulence shows that the ratio LM/LE ≈ 1 for weakly stratified flows. We show that in the limit of zero stratification, the turbulent Prandtl number is equal to the inverse of the ratio of the mechanical time scale to the scalar time scale, TL/Tρ. We use the stably stratified DNS data of Shih et al. (J. Fluid Mech., vol. 412, 2000, pp. 1–20; J. Fluid Mech., vol. 525, 2005, pp. 193–214) to propose a new parameterization for Prt in terms of the gradient Richardson number Ri. The formulation presented here provides a general framework for calculating Prt that will be useful for turbulence closure schemes in numerical models.

Instabilities of convection rolls in a high Prandtl number fluid

1971 ◽  
Vol 47 (2) ◽  
pp. 305-320 ◽  
Author(s):  
F. H. Busse ◽  
J. A. Whitehead
Keyword(s):  
Prandtl Number ◽  
Three Dimensional ◽  
Wave Number ◽  
Stationary Convection ◽  
Wave Numbers ◽  
Theoretical Predictions ◽  
High Prandtl Number ◽  
The Stability ◽  
Convection Rolls ◽  
Rayleigh Numbers

An experiment on the stability of convection rolls with varying wave-number is described in extension of the earlier work by Chen & Whitehead (1968). The results agree with the theoretical predictions by Busse (1967a) and show two distinct types of instability in the form of non-oscillatory disturbances. The ‘zigzag instability’ corresponds to a bending of the original rolls; in the ‘cross-roll instability’ rolls emerge at right angles to the original rolls. At Rayleigh numbers above 23,000 rolls are unstable for all wave-numbers and are replaced by a three-dimensional form of stationary convection for which the name ‘bimodal convection’ is proposed.

Eddy Diffusivity Based Comparisons of Turbulent Prandtl Number for Boundary Layer and Free Jet Flows With Reference to Fluids of Very Low Prandtl Number

1993 ◽  
Vol 115 (3) ◽  
pp. 549-552 ◽  
Author(s):  
K. Bremhorst ◽  
L. Krebs
Keyword(s):  
Boundary Layer ◽  
Prandtl Number ◽  
Eddy Diffusivity ◽  
Liquid Sodium ◽  
Jet Flows ◽  
Turbulent Prandtl Number ◽  
Free Jet ◽  
Functional Relationships ◽  
Prandtl Numbers ◽  
Universal Expression

It is shown that turbulent Prandtl numbers for turbulent boundary layer and jet flows correlate well if compared on the basis of eddy diffusivity. Experimental data for liquid sodium (Pr= 0.0058) lead to a universal expression relating turbulent Prandtl number to eddy diffusivity of heat with the same expression applying to boundary layer and jet flows. This suggests that the length and velocity scale dependence of the turbulent Prandtl number is associated predominantly with the eddy diffusivity of momentum. Since turbulent flow codes generally include eddy diffusivity of momentum calculations, simple and accurate estimates of eddy diffusivity of heat are possible without reliance on a variety of turbulent Prandtl number functional relationships for prediction of the temperature field. Available results also indicate that turbulent Prandtl number can be set at 0.9–1.0 irrespective of molecular Prandtl number, provided that εM/ν > 3Pr−1 and the structures of the mean temperature gradient and mean shear are similar.

Polydomain Structure of Epitaxial PbTiO3 films on MgO

1997 ◽  
Vol 493 ◽  
Author(s):  
S. P. Alpay ◽  
A. S. Prakash ◽  
S. Aggarwal ◽  
R. Ramesh ◽  
A. L. Roytburd ◽  
...  
Keyword(s):  
Domain Structure ◽  
Pulsed Laser ◽  
Laser Deposition ◽  
X Ray Diffraction ◽  
X Ray ◽  
Theoretical Predictions ◽  
The Stability ◽  
Increasing Temperature ◽  
Stable Domains ◽  
Domain Stability

ABSTRACTA PbTiO3(001) film grown on MgO(001) by pulsed laser deposition is examined as an example to demonstrate the applications of the domain stability map for epitaxial perovskite films which shows regions of stable domains and fractions of domains in a polydomain structure. X-ray diffraction studies indicate that the film has a …c/a/c/a… domain structure in a temperature range of °C to 400°C with the fraction of c-domains decreasing with increasing temperature. These experimental results are in excellent agreement with theoretical predictions based on the stability map.

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