## Computational ﬂuid dynamics in room air ﬂow analysis

Taking U,V andW to be the velocity components in the x, y and z directions respectively, ρ the ﬂuid density and t the time, then the rate of increase in the density ρ within the control volume dx dy dz equals the net rate of inﬂux of mass to the control volume, namely:

∂ρ

∂t + ∂ ∂x

(ρU)+ ∂ ∂y

(ρV)+ ∂ ∂z (ρW) = 0 (8.1)

If a turbulent ﬂow is considered (as is the case in most room ﬂow problems) and the velocities in equation (8.1) are replaced by the sum of a time-mean component and a ﬂuctuating component, i.e.:

U = u+ u′ V = v + v′ W = w +w′

the mass conservation equation becomes:

∂ρ

∂t + ∂ ∂x

(ρu)+ ∂ ∂y

(ρv)+ ∂ ∂z (ρw) = 0 (8.2)

In arriving at this equation it has been assumed that the ﬂuctuations in u′, v′ and w′ occur over a much shorter time interval than ∂t so that u ≈ U, v ≈ V and w ≈ W during this time interval. The density in equation (8.2) or the transport equations given in the succeeding sections may be represented by an equation of state, i.e. ρ = f (p,T).