## Table 3 Governing Differential Equations

$$ \frac{\partial\left(\rho\right)}{\partial t}+\frac{\partial\left(\rho\mathbf{u}\right)}{\partial\mathbf{x}}+\frac{\partial\left(\rho\mathbf{v}\right)}{\partial\mathbf{y}}+\frac{\partial\left(\rho\mathbf{w}\right)}{\partial\mathbf{z}}=0 $$

$$ \frac{\partial\left(\rho\mathrm{u}\right)}{\partial\mathrm{t}}+\frac{\partial\left(\rho\mathrm{u}^{2}-\tau_{\mathrm{X X}}\right)}{\partial\mathrm{x}}+\frac{\partial\left(\rho\mathrm{u v}-\tau_{\mathrm{X y}}\right)}{\partial\mathrm{y}}+\frac{\partial\left(\rho\mathrm{u w}-\tau_{\mathrm{X z}}\right)}{\partial\mathrm{z}}=--\frac{\partial\mathrm{P}}{\partial\mathrm{x}} $$

$$ \frac{\partial\left(\rho\mathrm{v}\right)}{\partial\mathrm{t}}+\frac{\partial\left(\rho\mathrm{u v}-\tau_{\mathrm{y x}}\right)}{\partial\mathrm{x}}+\frac{\partial\left(\rho\mathrm{v}^{2}-\tau_{\mathrm{y y}}\right)}{\partial\mathrm{y}}+\frac{\partial\left(\rho\mathrm{v w}-\tau_{\mathrm{y z}}\right)}{\partial\mathrm{z}}=-\frac{\partial\mathrm{P}}{\partial\mathrm{y}}-(\rho-\rho_{0})/\mathrm{F r} $$

$$ \frac{\partial\left(\rho\mathrm{w}\right)}{\partial\mathrm{t}}+\frac{\partial\left(\rho\mathrm{u w}-\tau_{\mathrm{z x}}\right)}{\partial\mathrm{x}}+\frac{\partial\left(\rho\mathrm{v w}-\tau_{\mathrm{z y}}\right)}{\partial\mathrm{y}}+\frac{\partial\left(\rho\mathrm{w}^{2}-\tau_{\mathrm{z z}}\right)}{\partial\mathrm{z}}=--\frac{g\mathrm{P}}{\partial\mathrm{z}} $$

$$ \frac{\partial\left(\rho\mathrm{T}\right)}{\partial\mathrm{t}}+\frac{\partial}{\partial\mathrm{x}}\left(\rho\mathrm{uT}-\left(\frac{1}{\mathrm{Re}\mathrm{Pr}}\frac{\partial\mathrm{T}}{\partial\mathrm{x}}\right)\right)+\frac{\partial}{\partial\mathrm{y}}\left(\rho\mathrm{vT}-\left(\frac{1}{\mathrm{Re}\mathrm{Pr}}\frac{\partial\mathrm{T}}{\partial\mathrm{y}}\right)\right)+\frac{\partial}{\partial\mathrm{z}}\left(\rho\mathrm{wT}-\left(\frac{1}{\mathrm{Re}\mathrm{Pr}}\frac{\partial\mathrm{T}}{\partial\mathrm{z}}\right)\right)=\mathrm{Q} $$

$$ \rho\mathrm{\bf T}=\mathrm{\bf P}/C_{1}+P_{0} $$

$$ \rho_{\mathrm{O}}=\exp{[-\mathrm{Y}/\mathrm{C}_{2}]} $$

$$ \tau_{\mathbf{x x}}\;=\;\frac{2}{3}\;\mu\left(2\;\frac{\partial\mathbf{u}}{\partial\mathbf{x}}\;-\;\frac{\partial\mathbf{v}}{\partial\mathbf{y}}\;-\;\frac{\partial\mathbf{w}}{\partial\mathbf{z}}\right) $$

$$ \tau_{\mathbf{x y}}=\mu(\frac{\partial\mathbf{u}}{\partial\mathbf{y}}+\frac{\partial\mathbf{v}}{\partial\mathbf{x}})=\tau_{\mathbf{y x}} $$

$$ \tau_{\mathbf{y y}}\;=\;\frac{2}{3}\mu\left(2\frac{\partial\mathbf{v}}{\partial\mathbf{y}}\;-\;\frac{\partial\mathbf{w}}{\partial\mathbf{z}}\;-\;\frac{\partial\mathbf{u}}{\partial\mathbf{x}}\right) $$

$$ \tau_{\mathbf{z}\mathbf{x}}=\mu(\frac{\partial\mathbf{w}}{\partial\mathbf{x}}+\frac{\partial\mathbf{u}}{\partial\mathbf{z}})=\tau_{\mathbf{x}\mathbf{z}} $$

$$ \tau_{z z}=\frac{2}{3}\mu(2\frac{\partial\mathbf{w}}{\partial z}-\frac{\partial\mathbf{u}}{\partial\mathbf{x}}-\frac{\partial\mathbf{v}}{\partial\mathbf{y}}) $$

$$ \tau_{\mathbf{y}\mathbf{z}}=\mu(\frac{\partial\mathbf{v}}{\partial\mathbf{z}}+\frac{\partial\mathbf{w}}{\partial\mathbf{y}})=\tau_{\mathbf{z}\mathbf{y}} $$

## NOMENCLATURE

$$ C_{1}\qquad:\quad\mathrm{c o n s t a n t}\;(\mathrm{=R_{0}T_{0}/u_{0}}^{2}) $$

$$ \mathrm{C_{2}\quad:\quad constant\quad(=\mathrm{R_{0}T_{0}}/\quad(gH))} $$

$ C_{p} $ : specific heat

g : gravitational acceleration

$$ \mathrm{F}_{\mathrm{r}}^{\mathrm{r}}:\mathrm{constant}\left(\mathrm{=u}_{\mathrm{O}}^{2}/\left(\mathrm{gH}\right)\right) $$

H : height of ceiling

k : thermal conductivity

P : pressure

Pr : Prandtle number (=μeffCp/k eff)

$ R_{e} $ : local Reynolds number

Q : heat release rate

$ R_{0} $ : gas constant

$$ (=\rho_{\mathbf{o}}\mathbf{u}_{\mathbf{o}}\mathbf{H}/\mathbf{\mu}_{\mathbf{e f f}}) $$

t : time

T : temperature

Y : height above floor

u,v,w : velocity components

$ \mu $ : viscosity

x,y,z : Cartesian coordinates

τ : viscous stress tensor

$ \rho $ : density

## (Subscripts)

eff : effective (laminar plus turbulent)

o : reference quantity