Empirical scaling laws for wall-bounded turbulence deduced from direct numerical simulations

2021 ◽  
Vol 6 (4) ◽  
Author(s):  
Philippe R. Spalart ◽  
Hiroyuki Abe
Keyword(s):  
Numerical Simulations ◽  
Scaling Laws ◽  
Direct Numerical Simulations ◽  
Wall Bounded Turbulence

scholarly journals Mixing Driven by Radiative and Evaporative Cooling at the Stratocumulus Top

2015 ◽  
Vol 72 (12) ◽  
pp. 4681-4700 ◽  
Author(s):  
Alberto de Lozar ◽  
Juan Pedro Mellado
Keyword(s):  
Numerical Simulations ◽  
Scaling Laws ◽  
Scaling Law ◽  
Evaporative Cooling ◽  
Direct Numerical Simulations ◽  
Radiative Cooling ◽  
Grid Spacing ◽  
Inversion Point ◽  
Entrainment Velocity ◽  
Diffusive Effects

Abstract The stratocumulus-top mixing process is investigated using direct numerical simulations of a shear-free cloud-top mixing layer driven by evaporative and radiative cooling. An extension of previous linear formulations allows for quantifying radiative cooling, evaporative cooling, and the diffusive effects that artificially enhance mixing and evaporative cooling in high-viscosity direct numerical simulations (DNS) and many atmospheric simulations. The diffusive cooling accounts for 20% of the total evaporative cooling for the highest resolution (grid spacing ~14 cm), but this can be much larger (~100%) for lower resolutions that are commonly used in large-eddy simulations (grid spacing ~5 m). This result implies that the κ scaling for cloud cover might be strongly influenced by diffusive effects. Furthermore, the definition of the inversion point as the point of neutral buoyancy allows the derivation of two scaling laws. The in-cloud scaling law relates the velocity and buoyancy integral scales to a buoyancy flux defined by the inversion point. The entrainment-zone scaling law provides a relationship between the entrainment velocity and the liquid evaporation rate. By using this inversion point, it is shown that the radiative-cooling contribution to the entrainment velocity decouples from the evaporative-cooling contribution and behaves very similarly as in the smoke cloud. Finally, evaporative and radiative cooling have similar strengths, when this strength is measured by the integrated buoyancy source. This result partially explains why current entrainment parameterizations are not accurate enough, given that most of them implicitly assume that only one of the two mechanisms rules the entrainment.

scholarly journals Scaling analysis and simulation of strongly stratified turbulent flows

2007 ◽  
Vol 585 ◽  
pp. 343-368 ◽  
Author(s):  
G. BRETHOUWER ◽  
P. BILLANT ◽  
E. LINDBORG ◽  
J.-M. CHOMAZ
Keyword(s):  
Reynolds Number ◽  
Numerical Simulations ◽  
Turbulent Flows ◽  
Scaling Laws ◽  
Three Dimensional ◽  
Direct Numerical Simulations ◽  
Small Scale ◽  
Scaling Analysis ◽  
Length Scale ◽  
Stratified Turbulence

Direct numerical simulations of stably and strongly stratified turbulent flows with Reynolds number Re ≫ 1 and horizontal Froude number Fh ≪ 1 are presented. The results are interpreted on the basis of a scaling analysis of the governing equations. The analysis suggests that there are two different strongly stratified regimes according to the parameter $\mathcal{R} \,{=}\, \hbox{\it Re} F^2_h$. When $\mathcal{R} \,{\gg}\, 1$, viscous forces are unimportant and lv scales as lv ∼ U/N (U is a characteristic horizontal velocity and N is the Brunt–Väisälä frequency) so that the dynamics of the flow is inherently three-dimensional but strongly anisotropic. When $\mathcal{R} \,{\ll}\, 1$, vertical viscous shearing is important so that $l_v \,{\sim}\, l_h/\hbox{\it Re}^{1/2}$ (lh is a characteristic horizontal length scale). The parameter $\cal R$ is further shown to be related to the buoyancy Reynolds number and proportional to (lO/η)4/3, where lO is the Ozmidov length scale and η the Kolmogorov length scale. This implies that there are simultaneously two distinct ranges in strongly stratified turbulence when $\mathcal{R} \,{\gg}\, 1$: the scales larger than lO are strongly influenced by the stratification while those between lO and η are weakly affected by stratification. The direct numerical simulations with forced large-scale horizontal two-dimensional motions and uniform stratification cover a wide Re and Fh range and support the main parameter controlling strongly stratified turbulence being $\cal R$. The numerical results are in good agreement with the scaling laws for the vertical length scale. Thin horizontal layers are observed independently of the value of $\cal R$ but they tend to be smooth for $\cal R$< 1, while for $\cal R$ > 1 small-scale three-dimensional turbulent disturbances are increasingly superimposed. The dissipation of kinetic energy is mostly due to vertical shearing for $\cal R$ < 1 but tends to isotropy as $\cal R$ increases above unity. When $\mathcal{R}$ < 1, the horizontal and vertical energy spectra are very steep while, when $\cal R$ > 1, the horizontal spectra of kinetic and potential energy exhibit an approximate k−5/3h-power-law range and a clear forward energy cascade is observed.

scholarly journals Triple Decomposition of Velocity Gradient Tensor in Compressible Turbulence

Fluids ◽  
2021 ◽  
Vol 6 (3) ◽  
pp. 98
Author(s):  
Radouan Boukharfane ◽  
Aimad Er-raiy ◽  
Linda Alzaben ◽  
Matteo Parsani
Keyword(s):  
Rigid Body ◽  
Numerical Simulations ◽  
Scaling Laws ◽  
Isotropic Turbulence ◽  
Three Dimensional ◽  
Direct Numerical Simulations ◽  
Compressible Turbulence ◽  
Local Motion ◽  
Eighth Order ◽  
Compressibility Effects

The decomposition of the local motion of a fluid into straining, shearing, and rigid-body rotation is examined in this work for a compressible isotropic turbulence by means of direct numerical simulations. The triple decomposition is closely associated with a basic reference frame (BRF), in which the extraction of the biasing effect of shear is maximized. In this study, a new computational and inexpensive procedure is proposed to identify the BRF for a three-dimensional flow field. In addition, the influence of compressibility effects on some statistical properties of the turbulent structures is addressed. The direct numerical simulations are carried out with a Reynolds number that is based on the Taylor micro-scale of Reλ=100 for various turbulent Mach numbers that range from Mat=0.12 to Mat=0.89. The DNS database is generated with an improved seventh-order accurate weighted essentially non-oscillatory scheme to discretize the non-linear advective terms, and an eighth-order accurate centered finite difference scheme is retained for the diffusive terms. One of the major findings of this analysis is that regions featuring strong rigid-body rotations or straining motions are highly spatially intermittent, while most of the flow regions exhibit moderately strong shearing motions in the absence of rigid-body rotations and straining motions. The majority of compressibility effects can be estimated if the scaling laws in the case of compressible turbulence are rescaled by only considering the solenoidal contributions.

Prandtl number dependence of Nusselt number in direct numerical simulations

2000 ◽  
Vol 419 ◽  
pp. 325-344 ◽  
Author(s):  
ROBERT M. KERR ◽  
JACKSON R. HERRING
Keyword(s):  
Experimental Data ◽  
Boundary Layer ◽  
Nusselt Number ◽  
Prandtl Number ◽  
Numerical Simulations ◽  
Scaling Laws ◽  
Direct Numerical Simulations ◽  
Nusselt Numbers

The dependence of the Nusselt number Nu on the Rayleigh Ra and Prandtl Pr number is determined for 104 < Ra < 107 and 0.07 < Pr < 7 using DNS with no-slip upper and lower boundaries and free-slip sidewalls in a 8 × 8 × 2 box. Nusselt numbers, velocity scales and boundary layer thicknesses are calculated. For Nu there are good comparisons with experimental data and scaling laws for all the cases, including Ra2/7 laws at Pr = 0.7 and Pr = 7 and at low Pr, a Ra1/4 regime. Calculations at Pr = 0.3 predict a new Nu ∼ Ra2/7 regime at slightly higher Ra than the Pr = 0.07 calculations reported here and the mercury Pr = 0.025 experiments.

Circulation deposition on shock-accelerated planar and curved density-stratified interfaces: models and scaling laws

1994 ◽  
Vol 269 ◽  
pp. 45-78 ◽  
Author(s):  
Ravi Samtaney ◽  
Norman J. Zabusky
Keyword(s):  
Numerical Simulations ◽  
Scaling Laws ◽  
Unit Length ◽  
Density Ratio ◽  
Bubble Surface ◽  
Incident Shock ◽  
Direct Numerical Simulations ◽  
Specific Heats ◽  
Gamma Prime ◽  
Total Circulation

Vorticity is deposited baroclinically by shock waves on density inhomogeneities. In two dimenslons, the circulation deposited on a planar interface may be derived analytically using shock polar analysis provided the shock refraction is regular. We present analytical expressions for Γ′, the circulation deposited per unit length of the unshocked planar interface, within and beyond the regular refraction regime. To lowest order, Γ′ scales as \[ \Gamma^\prime\propto (1-\eta^{-\frac{1}{2}})(\sin\alpha)(1+M^{-1}+2M^{-2})(M-1)(\gamma^{\frac{1}{2}}/\gamma + 1)\] where M is the Mach number of the incident shock, η is the density ratio of the gases across the interface, α is the angle between the shock and the interface and γ is the ratio of specific heats for both gases. For α ≤ 30°, the error in this approximation is less than 10% for 1.0 < M ≤ 1.32 for all η > 1, and 5.8 ≤ η ≤ 32.6 for all M. We validate our results by quantification of direct numerical simulations of the compressible Euler equations with a second-order Godunov code.We generalize the results for total circulation on non-planar (sinusoidal and circular) interfaces. For the circular bubble case, we introduce a ‘near-normality’ ansatz and obtain a model for total circulation on the bubble surface that agrees well with results of direct numerical simulations. A comparison with other models in the literature is presented.

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