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 4- 2approx.tex\hfil \today
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  \section{Approximations for small temperature variation} 
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\subsection{Mass conservation and almost incompressibilty}
Recall the law of mass conservation:
 \[ - \f{1}{\rho} \f{D\rho}{Dt} = \nabla \cdot \vec{q}\]
Let the time scale be $L/U$. The left-hand-side is of the order $ \f{U}{L} \f{\Delta \rho}{\rho}$ while the right-hand-side is $\f{U}{L} $.  For $ \Delta T =O(10^o C)$, their ratio is \[  \f{\Delta \rho}{\rho} \sim   \f{\Delta T}{T} \sim \f{10^o K}{300^o K} \ll 1\] 
Therefore,
\be \nabla \cdot \vec{q} = 0 . \label{continuity}\ee
The fluid is approximately incompressible even if $\Delta T \neq 0$.  
\subsection{Momentum conservation and Boussinesq approximation}
  In static equilibirum $\vec{q}_o \equiv 0$.  Therefore,
\be -\nabla p_o + \vec{f} \rho_o = 0 . \ee
Let $p = p_d + p_o$ where $ p_d  $ is the  dynamic pressure
\[ \rho = \rho_d + \rho_o \]
\[ -\nabla p + \rho \vec{f} = -\nabla p_o + \rho_o \vec{f} - \nabla p_d + \lp \rho - \rho_o \rp \vec{f} \]
Therefore,
\be \rho \f{D\vec{q}}{Dt} = -\nabla p_d + \nabla \cdot \stackrel{=}{\tau} + \underbrace{\lp \rho - \rho_o \rp \vec{f}}_{\mbox{buoyancy force}} \ee
Now \be \rho = \bar\rho_o[1-\beta(\Delta T_o+\Delta T_d)])\ee
Hence \[ \rho_o = \bar\rho_o(1-\beta \Delta T_o), ~~ \rho_d = -\bar\rho_o\beta \Delta T_d , \]
and \be (\rho-\rho_o) \vec f = -\bar \rho_o(-g) \beta \Delta T_d \vec k= \bar \rho_o g \beta \Delta T_d \vec k\ee
For mildly varying $\rho_o$ and small $\rho - \rho_o$, we ignore the variation of density and approximate $\rho_o$ by a constant everywhere, except in the body force.  This is called the {\bf Boussinesq approximation}. Thus 
\be \bar{\rho}_o \f{D\vec q}{Dt} = -\nabla p_d + \nabla \cdot \stackrel{=}{\tau} + \bar{\rho}_o g\beta \Delta T_d \, \vec{k} \ee
where
\[ \bar{\rho}_o = \rho_o (z = 0 ) \]

\subsection{Total energy}    Using Eqn. (\ref{continuity}) in Eqn. (\ref{temp}) and the Boussinesq approximation
\be \bar{\rho_o} C \f{DT}{Dt} = \f{\p}{\p x_i} K \f{\p T}{\p x_i} + \Phi \ee
Here $T$ is the total temperature (static + dynamic). 

Now
\[ \f{\Phi}{\bar\rho_o C \f{DT}{Dt}}\sim \f{\mu U^2/L^2}{\bar\rho_o C\f{U\Delta T}{L}}\sim \f{\mu}{\bar\rho_oUL}\f{U^2}{C\Delta T}=\f{E}{Re}\]
where \[
E=\f{U^2}{C\Delta T} = \mbox{Eckart No.}, ~~ Re= \f{\rho UL}{\mu} =\mbox{Reynolds No.}\]
\begin{table}[h]\caption{Typical values $E/Re$ for air and water}  \begin{center}\begin{tabular}{|c|c|c|} 
\hline
 & Water & Air\\  \hline C (erg/s-gr-${}^\circ K$) &$4\times 10^7$ & $10^7$\\ \hline 
K (ergs-cm-${}^\circ K$) & $0.6\times 10^5$ & $0.3\times 10^5$\\ \hline 
$\nu (cm^2/s)$ & $10^2$ & $2\times 10^{-2}$\\ \hline
$\beta (1/^\circ K)$ & $10^{-3}$ & 1/300 \\ \hline 
$E  $& $0.25\times 10^{-2}$ & $10^{-4}$\\
\hline  
$Re$ & $10^5$ & $ 0.5 \times 10^5$ \\
\hline 
\end{tabular}\end{center}\label{table:eckart}\end{table}
In environmental problems,  $\Delta T \sim 10~^oK, L \sim 10~m, U \sim 1 m/sec$, the last two columns of Table \ref{table:eckart} is obtained. 
 Hence $\Phi$ is negligible,
and
\be \bar{\rho}_o C \f{DT}{Dt} = \f{\p}{\p x_i} K \f{\p T}{\p x_i}\label{heat} \ee
Only convection and diffusion are dominant. This is typical in natural convectin problems. 

{\bf Remark 1.} In many engineering problems (aerodynamics, rocket reentry, etc.), 
  heat is caused by frictional dissipation in the flow, therefore, $\Phi$ is important. These are called {\it  forced convection} problems. In environmental problems,  flow is often the result of heat addition. Here the  flow problems are referred to as  the {\it natural convection}. 
 
 {\bf Remark 2}: Since $\bar{T}$ appears as a derivative only, only the variation of $T$, i.e.,   the difference $T - \bar{T}_o$ matters, where $\bar{T}_o$ is a reference temperature.

 {\bf Remark 3}: In turbulent natural convection
\be u = \bar{u} + u' \qquad T = \bar{T} + T' \ee
Averaging Eqn. (\ref{heat})
\be \bar{\rho}_o c \f{D\bar{T}}{Dt} = - \underbrace{\bar{\rho}_o c \f{\p}{\p x_i} \overline{u'_iT'}}_{\mbox{heat flux by turbulence}} + \f{\p}{\p x_i} K \f{\p \bar{T}}{\p x_i} \ee
If the the correlation term is modeled as eddy diffusion, the form would be similar to (\ref{heat}).


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