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Investigations On High Rayleigh Number Turbulent Free Convection

by Puthenveettil, Baburaj A

Abstract (Summary)
High Rayleigh number(Ra) turbulent free convection has many unresolved

issues related to the phenomenology behind the flux scaling, the

presence of a mean wind and its effects, exponential probability

distribution functions, the Prandtl number dependence and the nature

of near wall structures. Few studies have been conducted in the high

Prandtl number regime and the understanding of near wall coherent

structures is inadequate for $Ra > 10^9$. The present thesis deals

with the results of investigations conducted on high Rayleigh

number turbulent free convection in the high Schmidt number(Sc)

regime, focusing on the role of near wall coherent structures.

We use a new method of driving the convection using concentration

difference of NaCl across a horizontal membrane between two tanks to

achieve high Ra utilising the low molecular diffusivity of NaCl. The

near wall structures are visualised by planar laser induced

fluorescence. Flux is estimated from transient measurement of

concentration in the top tank by a conductivity probe. Experiments

are conducted in tanks of $15\times15\times 23$cm (aspect ratio,AR =

0.65) and $10\times10\times 23$cm (AR = 0.435). Two membranes of

0.45$\mu$ and 35$\mu$ mean pore size were used. For the fine

membrane (and for the coarse membrane at low driving potentials), the

transport across the partition becomes diffusion dominated, while the

transport above and below the partition becomes similar to unsteady

non penetrative turbulent free convection above flat horizontal

surfaces (Figure~\ref{fig:schem}(A)). In this type of convection,

the flux scaled as $q\sim \Delta C_w ^{4/3}$,where $\Delta C_w$ is

the near wall concentration difference, similar to that in Rayleigh -

B\'{e}nard convection . Hence, we are able to study turbulent free

convection over horizontal surfaces in the Rayleigh Number range of

$\sim 10^{10}- 10 ^{11}$ at Schmidt number of 602, focusing on the

nature and role of near wall coherent structures. To our knowledge,

this is the first study showing clear images of near wall structures

in high Rayleigh Number - high Schmidt number turbulent free

convection.

We observe a weak flow across the membrane in the case of the coarser

membrane at higher driving potentials (Figure \ref{fig:schem}(B)).

The effect of this through flow on the flux and the near wall

structures is also investigated. In both the types of convection the

near wall structure shows patterns formed by sheet plumes, the common

properties of these patterns are also investigated. The major

outcomes in the above three areas of the thesis can be summarised as

follows

\subsection*{Primary findings}

\label{sec:primary-findings}

\subsubsection*{High Ra - Sc turbulent free convection }

\label{sec:-hrc-}

The non-dimensional flux was similar to that reported by

Goldstein\cite{gcs} at Sc of 2750. Visualisations show that the near

wall coherent structures are line plumes. Depending on the Rayleigh

number and the Aspect ratio, different types of large scale flow cells

which are driven by plume columns are observed. Multiple large scale

flow cells are observed for AR = 0.65 and a single large scale flow

for AR= 0.435. The large scale flow create a near wall mean shear,

which is seen to vary across the cross section. The orientation of the

large scale flow is seen to change at a time scale much larger than

the time scale of one large scale circulation

The near wall structures show interaction of the large scale flow with

the line plumes. The plumes are initiated as points and then gets

elongated along the mean shear direction in areas of larger mean

shear. In areas of low mean shear, the plumes are initiated as points

but gets elongated in directions decided by the flow induced by the

adjacent plumes. The effect of near wall mean shear is to align the

plumes and reduce their lateral movement and merging. The time scale

for the merger of the near wall line plumes is an order smaller than

the time scale of the one large scale circulation. With increase in

Rayleigh number, plumes become more closely and regularly spaced.

We propose that the near wall boundary layers in high Rayleigh number

turbulent free convection are laminar natural convection boundary

layers. The above proposition is verified by a near wall model,

similar to the one proposed by \cite{tjfm}, based on the similarity

solutions of laminar natural convection boundary layer equations as

Pr$\rightarrow\infty$. The model prediction of the non dimensional

mean plume spacing $Ra_\lambda^{1/3}~=~\lambda /Z_w~=~91.7$ - where

$Ra_\lambda$ is the Rayleigh number based on the plume spacing

$\lambda$, and $Z_w$ is a near wall length scale for turbulent free

convection - matches the experimental measurements. Therefore, higher

driving potentials, resulting in higher flux, give rise to lower mean

plume spacing so that $\lambda \Delta C_w^{1/3}$ or $\lambda q^{1/4}$ is

a constant for a given fluid.

We also show that the laminar boundary layer assumption is consistent

with the flux scaling obtained from integral relations. Integral

equations for the Nusselt number(Nu) from the scalar variance

equations for unsteady non penetrative convection are derived.

Estimating the boundary layer dissipation using laminar natural

convection boundary layers and using the mean plume spacing relation,

we obtain $Nu\sim Ra^{1/3}$ when the boundary layer scalar dissipation

is only considered. The contribution of bulk dissipation is found to

be a small perturbation on the dominant 1/3 scaling, the effect of

which is to reduce the effective scaling exponent.

In the appendix to the thesis, continuing the above line of reasoning,

we conduct an exploratory re-analysis (for $Pr\sim 1$) of the Grossman

and Lohse's\cite{gl} scaling theory for turbulent Rayleigh - B\'enard

convection. We replace the Blasius boundary layer assumption of the

theory with a pair of externally forced laminar natural convection

boundary layers per plume. Integral equations of the externally forced

laminar natural convection boundary layer show that the mixed

convection boundary layer thickness is decided by a $5^{th}$ order

algebraic equation, which asymptotes to the laminar natural convection

boundary layer for zero mean wind and to Blasius boundary layer at

large mean winds.

\subsubsection*{Effect of wall normal flow on flux and near wall structures}

\label{sec:effect-wall-normal}

For experiments with the coarser($35\mu$) membrane, we observe three

regimes viz. the strong through flow regime

(Figure~\ref{fig:schem}(b)), the diffusion regime (Figure

\ref{fig:schem}(a)), and a transition regime between the above two

regimes that we term as the weak through flow regime.

At higher driving potentials, only half the area above the coarser

membrane is covered by plumes, with the other half having plumes below

the membrane. A wall normal through flow driven by impingement of the

large scale flow is inferred to be the cause of this (Figure

\ref{fig:schem}(b)). In this strong through flow regime, only a single

large scale flow circulation cell oriented along the diagonal or

parallel to the walls is detected. The plume structure is more

dendritic than the no through flow case. The flux scales as $\Delta

C_w^n$, with $7/3\leq n\leq 3$ and is about four times that observed

with the fine membrane. The phenomenology of a flow across the

membrane driven by the impingement of the large scale flow of strength

$W_*$, the Deardorff velocity scale, explains the cubic scaling. We

find the surprising result that the non-dimensional flux is smaller

than that in the no through flow case for similar parameters.

The mean plume spacings in the strong through flow regime are larger

and show a different Rayleigh number dependence vis-a-vis the no

through flow case. Using integral analysis, an expression for the

boundary layer thickness is derived for high Schmidt number laminar

natural convection boundary layer with a normal velocity at the wall.

(Also, solutions to the integral equations are obtained for the

$Sc\sim 1$ case, which are given as an Appendix.) Assuming the

gravitational stability condition to hold true, we show that the plume

spacing in the high Schmidt number strong through flow regime is

proportional to $\sqrt{Z_w\,Z{_{v_i}}}$, where $Z{_{v_i}}$ is a length

scale from the through flow velocity. This inference is fairly

supported by the plume spacing measurements

At lower driving potentials corresponding to the transition regime,

the whole membrane surface is seen to be covered by plumes and the

flux scaled as $\Delta C_w^{4/3}$.

The non-dimensional flux is about the same as in turbulent free

convection over flat surfaces if $\frac{1}{2}\Delta C $ is assumed to

occur on one side of the membrane. This is expected to occur in the

area averaged sense with different parts of the membrane having

predominance of diffusion or through flow dominant transport. At very

low driving potentials corresponding to the diffusion regime, the

diffusion corrected non dimensional flux match the turbulent free

convection values, implying a similar phenomena as in the fine

membrane.

\subsubsection*{Universal probability distribution of near wall structures}

\label{sec:univ-prob-distr}

We discover that the probability distribution function of the plume

spacings show a standard log normal distribution, invariant of the

presence or the absence of wall normal through flow and at all the

Rayleigh numbers and aspect ratios investigated. These plume

structures showed the same underlying multifractal spectrum of

singularities in all these cases. As the multifractal curve indirectly represents the processes by which

these structures are formed, we conclude that the plume structures are created by a common

generating mechanism involving nucleation at points, growth along

lines and then merging, influenced by the external mean shear.

Inferring from the thermodynamic analogy of multifractal analysis, we

hypothesise that the near wall plume structure in turbulent free

convection might be formed so that the entropy of the structure is

maximised within the given constraints.

Bibliographical Information:

Advisor:Arakeri, Jaywant

School:Indian Institute of Science

School Location:India

Source Type:Master's Thesis

Keywords:fluid and plasma physics convection heat rayleigh number plumes mean wind near wall dynamics turbulent natural structures benard

ISBN:

Date of Publication:06/01/2004

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