Statistical Approach to Inhomogeneous Turbulent Diffusion: General Formulation and Diffusion of a Passive Scalar in Wall Turbulence

AbstractWe perform experiments to study the mixing of passive scalar by a buoyancy-induced turbulent flow in a long narrow vertical tank. The turbulent flow is associated with the downward mixing of a small flux of dense aqueous saline solution into a relatively large upward flux of fresh water. In steady state, the mixing region is of finite extent, and the intensity of the buoyancy-driven mixing is described by a spatially varying turbulent diffusion coefficient $\kappa _v(z)$ which decreases linearly with distance $z$ from the top of the tank. We release a pulse of passive scalar into either the fresh water at the base of the tank, or the saline solution at the top of the tank, and we measure the subsequent mixing of the passive scalar by the flow using image analysis. In both cases, the mixing of the passive scalar (the dye) is well-described by an advection–diffusion equation, using the same turbulent diffusion coefficient $\kappa _v(z)$ associated with the buoyancy-driven mixing of the dynamic scalar. Using this advection–diffusion equation with spatially varying turbulent diffusion coefficient $\kappa _v(z)$, we calculate the residence time distribution (RTD) of a unit mass of passive scalar released as a pulse at the bottom of the tank. The variance in this RTD is equivalent to that produced by a uniform eddy diffusion coefficient with value $\kappa _e= 0.88 \langle \kappa _v \rangle $, where $\langle \kappa _v \rangle $ is the vertically averaged eddy diffusivity. The structure of the RTD is also qualitatively different from that produced by a flow with uniform eddy diffusion coefficient. The RTD using $\kappa _v$ has a larger peak value and smaller values at early times, associated with the reduced diffusivity at the bottom of the tank, and manifested mathematically by a skewness $\gamma _1\approx 1.60$ and an excess kurtosis $\gamma _2\approx 4.19 $ compared to the skewness and excess kurtosis of $\gamma _1\approx 1.46$, $\gamma _2 \approx 3.50$ of the RTD produced by a constant eddy diffusion coefficient with the same variance.

Download Full-text

Horizontal turbulent diffusion in a convective mixed layer

Journal of Fluid Mechanics ◽

10.1017/jfm.2014.545 ◽

2014 ◽

Vol 758 ◽

pp. 553-564 ◽

Cited By ~ 5

Author(s):

Junshi Ito ◽

Hiroshi Niino ◽

Mikio Nakanishi

Keyword(s):

Scalar Field ◽

Mixed Layer ◽

Turbulent Diffusion ◽

Passive Scalar ◽

Rate Of Change ◽

Horizontal Gradient ◽

Scalar Flux ◽

Turbulent Diffusion Coefficient ◽

Convective Mixed Layer ◽

Convergence Zones

AbstractA large eddy simulation (LES) is used to estimate a reliable horizontal turbulent diffusion coefficient, $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}K_{{h}}$, in a convective mixed layer (CML). The introduction of a passive scalar field with a fixed horizontal gradient at a given time enables $K_{{h}}$ estimation as a function of height, based on the simulated turbulent horizontal scalar flux. Here $K_{{h}}$ is found to be of the order of $100\ {\mathrm{m}}^2\ {\mathrm{s}}^{-1}$ for a typical terrestrial atmospheric CML. It is shown to scale by the product of the CML convective velocity, $w_{*}$, and its depth, $h$. Here $K_{{h}}$ is characterized by a vertical profile in the CML: it is large near both the bottom and top of the CML, where horizontal flows associated with convection are large. The equation pertaining to the temporal rate of change of a horizontal scalar flux suggests that $K_{{h}}$ is determined by a balance between production and pressure correlation at a fully developed stage. Pressure correlation near the bottom of the CML is localized in convergence zones near the boundaries of convective cells and becomes large within an eddy turnover time, $h/w_{*}$, after the introduction of the passive scalar field.

Download Full-text

The Structure of Turbulent Diffusion : 1st Report, Diffusion from a Source of Finite Width and Diffusion in a Mixing Region of Laminar and Turbulent Flows

Bulletin of JSME ◽

10.1299/jsme1958.21.1000 ◽

1978 ◽

Vol 21 (156) ◽

pp. 1000-1007

Author(s):

Masataka SHIRAKASHI ◽

Yukio TOMITA

Keyword(s):

Turbulent Flows ◽

Turbulent Diffusion ◽

Finite Width ◽

And Diffusion ◽

Laminar And Turbulent Flows

Download Full-text

THE EFFECT OF WIND ON CURRENTS AND DIFFUSION IN COASTAL SEA AREAS

Coastal Engineering Proceedings ◽

10.9753/icce.v14.140 ◽

1974 ◽

Vol 1 (14) ◽

pp. 140

Author(s):

B. Quetin

Keyword(s):

Velocity Profile ◽

Turbulent Flow ◽

Turbulent Diffusion ◽

Turbulent Viscosity ◽

Viscosity Coefficient ◽

Classic Theory ◽

Coastal Sea ◽

The Mediterranean ◽

And Diffusion

The calculation of turbulent flow using Navier's equations assumes the introduction of a turbulent viscosity coefficient the value of which is normally constant, conforming with Boussinesq's hypothesis. It was shown that setting aside this hypothesis, a velocity profile quite different to that resulting from the classic theory is obtained in the case of flow induced by wind. This result appears to be confirmed by the tests carried out in the Mediterranean. The advantage of this method is that it gives the vertical turbulent diffusion which is of particular interest to pollution studies.

Download Full-text

A hierarchical random additive model for passive scalars in wall-bounded flows at high Reynolds numbers

Journal of Fluid Mechanics ◽

10.1017/jfm.2018.139 ◽

2018 ◽

Vol 842 ◽

pp. 354-380 ◽

Cited By ~ 15

Author(s):

Xiang I. A. Yang ◽

Mahdi Abkar

Keyword(s):

Velocity Fluctuation ◽

Passive Scalar ◽

Streamwise Velocity ◽

Wall Turbulence ◽

Ensemble Averaging ◽

Outer Length ◽

Passive Scalars ◽

Streamwise Velocity Fluctuation ◽

Wall Bounded Flows ◽

High Reynolds

The kinematics of a fully developed passive scalar is modelled using the hierarchical random additive process (HRAP) formalism. Here, ‘a fully developed passive scalar’ refers to a scalar field whose instantaneous fluctuations are statistically stationary, and the ‘HRAP formalism’ is a recently proposed interpretation of the Townsend attached eddy hypothesis. The HRAP model was previously used to model the kinematics of velocity fluctuations in wall turbulence:$u=\sum _{i=1}^{N_{z}}a_{i}$, where the instantaneous streamwise velocity fluctuation at a generic wall-normal location$z$is modelled as a sum of additive contributions from wall-attached eddies ($a_{i}$) and the number of addends is$N_{z}\sim \log (\unicode[STIX]{x1D6FF}/z)$. The HRAP model admits generalized logarithmic scalings including$\langle \unicode[STIX]{x1D719}^{2}\rangle \sim \log (\unicode[STIX]{x1D6FF}/z)$,$\langle \unicode[STIX]{x1D719}(x)\unicode[STIX]{x1D719}(x+r_{x})\rangle \sim \log (\unicode[STIX]{x1D6FF}/r_{x})$,$\langle (\unicode[STIX]{x1D719}(x)-\unicode[STIX]{x1D719}(x+r_{x}))^{2}\rangle \sim \log (r_{x}/z)$, where$\unicode[STIX]{x1D719}$is the streamwise velocity fluctuation,$\unicode[STIX]{x1D6FF}$is an outer length scale,$r_{x}$is the two-point displacement in the streamwise direction and$\langle \cdot \rangle$denotes ensemble averaging. If the statistical behaviours of the streamwise velocity fluctuation and the fluctuation of a passive scalar are similar, we can expect first that the above mentioned scalings also exist for passive scalars (i.e. for$\unicode[STIX]{x1D719}$being fluctuations of scalar concentration) and second that the instantaneous fluctuations of a passive scalar can be modelled using the HRAP model as well. Such expectations are confirmed using large-eddy simulations. Hence the work here presents a framework for modelling scalar turbulence in high Reynolds number wall-bounded flows.

Download Full-text