Natural ventilation

Natural ventilation exchanges indoor and outdoor fluid through openings using pressure differences created by buoyancy or wind.

For an ideal two-layer natural ventilation transient following a heating change $Q\mapsto \lambda Q$, let $V_0$ and ${\theta }_0$ be the initial flow and temperature excess, $s=V_{0}t/(A_{r}H)$, $\eta =h/H$, and $x=\eta (1-\theta /{\theta }_0)$. The integrated buoyancy pressure gives $V/V_0=\sqrt{1-x}$, while heat balance gives $x_s=\sqrt{1-x}-\lambda$. A material interface satisfies ${\eta }_s=\sqrt{1-x}$, so $\eta =x+\lambda s$. For $0<\lambda <1$, $x$ increases toward $1-{\lambda }^2$ and ${\eta }_s\ge \lambda$, ensuring finite-time filling of the room. After filling, $y=\theta /{\theta }_0$ obeys $y_s=\lambda -y^{3/2}$ and tends to ${\lambda }^{2/3}$. This model assumes well-mixed individual layers, negligible wall heat capacity and a sharp advected interface.

For a distributed heat source, a well-mixed interior with temperature excess $\theta$ gains temperature-volume flux $\mathcal{H}$ and loses $|q|\theta$ by ventilation in either direction. With signed upward throughflow, $q|q|=K_A(G_T\theta \pm W)$, where $G_T=g{\alpha }_T{H}_b$ and $W$ is wind pressure divided by mass density. The plus sign is assisting wind. Steady temperatures satisfy ${\theta }^2(G_T\theta +W)={\mathcal{H}}^2/K_A$ for assistance, or ${\theta }^2|G_T\theta -W|={\mathcal{H}}^2/K_A$ for opposition. Direction restrictions must be applied to roots.

For $0<\epsilon <2/(3\sqrt{3})$, the well-mixed ventilation temperature balance has one stable upward-flow temperature and two downward-flow temperatures. The heat-removal curve on the latter interval is $\sqrt{K_A}\theta \sqrt{W-G_T\theta }$, maximal at $2W/(3G_T)$. Its lower-temperature root is stable and its higher-temperature root unstable. The two meet at a saddle-node bifurcation. The unstable root separates the basins of the stable wind-dominated and buoyancy-dominated states.

Increasing wind or reducing heating both reduce the control parameter of opposing-wind ventilation bistability. However, wind moves the reversal threshold ${\theta }_w=W/G_T$, whereas heating does not. A sufficiently abrupt wind increase can place a hot upward-flow state in the cooler state's basin. Reducing positive heating at fixed wind cannot make an upward-flow state cross ${\theta }_w$, because its temperature time derivative there is positive. Slow variation follows a persistent stable branch; a fold destroys the cooler branch when wind is weakened or heat input increased.

An effective opening area may absorb both the discharge coefficient and the factor of two in an orifice law. If $A_e=\sqrt{2}{C}_d{A}_{\mathrm{geom}}$, its discharge is $q=A_e\sqrt{\delta p/\rho }$. Two equal openings carrying the same flux share the total pressure drop, so $q^2=A_e^2\Delta p/(2\rho )$. They do not share equal drops when an internal volume source makes their fluxes unequal. State the effective-area convention before comparing numerical factors in natural ventilation models.

Displacement ventilation supplies relatively cool air at low level and removes warmer air at high level. In the ideal two-layer model, buoyant plumes feed a warm upper layer above a sharp density interface while the supply occupies the lower layer.

For a pure plume with $g_p^{\prime }=\gamma B^{2/3}{z}^{-5/3}$, a steady sharp interface at $z_i$ has $Q_p=B^{1/3}{z}_i^{5/3}/\gamma$ and warm-layer reduced gravity $g^{\prime }=B/Q_p$. A freely discharging upper doorway of height $D$ and width $W$ has available head $D-z_i$. Equating plume supply to the broad-crested weir flux gives the displayed implicit height relation, with $0<z_i<D$. Warm air above the doorway stores heat but adds no available head at its crest in this uniform-layer model.

Two independent point-source pure plumes with $\psi =B_1/B_2<1$ remove lower-layer volume $Q=C(B_1^{1/3}+B_2^{1/3})h_1^{5/3}$. If plume 1 feeds the middle layer, its reduced gravity is $g_1^{\prime }=B_1/[C{B}_1^{1/3}{h}_1^{5/3}]$. The upper outflow carries both sources, so $g_2^{\prime }=(B_1+B_2)/Q$, yielding the displayed ratio. Opening flow is driven by the integrated hydrostatic head $g_1^{\prime }(h_2-h_1)+g_2^{\prime }(H-h_2)$. A plume crossing the lower interface has altered buoyancy relative to its new ambient and must be continued with nonzero incoming mass and momentum fluxes rather than reset as a zero-flux pure plume.

A source volume $Q$ makes roof outflow exceed floor inflow by $Q$. At zero floor inflow, an entraining plume cannot support a steady cold lower layer; the warm layer reaches the floor. With reduced gravity $g^{\prime }=B/Q$, the entire hydrostatic head drives roof discharge $Q=A_e\sqrt{g^{\prime }H}$. This gives the stated threshold. Equal splitting of pressure head is inappropriate at blocking. The same result in physical-area notation is $Q_c=(2C_d^2{A}_{\mathrm{geom}}^{2}BH{)}^{1/3}$.

For total plume flux $P(h)$, fixed ceiling exhaust $F$, zero ambient buoyancy, and uniform upper-layer reduced gravity $b$, the ideal two-layer balances areAt $P(h_{\ast })=F$ and $b_{\ast }=B/F$, linearization yields the decay times ${\tau }_h=A/P^{\prime }(h_{\ast })$ and ${\tau }_b=A(H-h_{\ast })/F$. For $P(h)\propto h^{5/3}$, ${\tau }_h=3A{h}_{\ast }/(5F)$. Upper-layer buoyancy relaxation usually dominates when $h_{\ast }\ll H$. Exact equilibrium is approached asymptotically; a fixed small relative tolerance takes order $\max ({\tau }_h,{\tau }_b)\log (1/\epsilon )$ after the initial transient.

Consider entraining plumes supplied with total source volume flux $S$, a separate lower-layer ambient inflow $L$, and ceiling exhaust $F=S+L$. Write $P(h)=S+E(h)$, where $E(h)$ is total fluid entrainment from the lower layer. Volume conservation gives $A\dot{h}=L-E(h)$. A steady positive lower layer therefore requires $L=E(h)>0$. If $L=0$ and $E(h)>0$ for every $h>0$, no positive steady displacement-ventilation interface height exists. In a shifted pure plume approximation $P_i(h)=\lambda B_i^{1/3}(h+z_i{)}^{5/3}$ matched to source flux $S_i$ at $h=0$, this follows immediately from $P_i(h)>S_i$ for $h>0$. This is a statement about the ideal two-layer model, not a prohibition of all possible stratification in a real ventilated room.

Suppose $n$ independent axisymmetric pure plumes share total buoyancy flux $B$ equally, have negligible source volume flux, and are supplied with a separate ambient ventilation flux $F$ at floor level. Their combined flux is $P(h)=\lambda n^{2/3}{B}^{1/3}{h}^{5/3}$. Volume conservation at a steady displacement-ventilation interface height and buoyancy flux conservation giveA positive two-layer state needs $0<h_{\ast }<H$, sources small relative to $h_{\ast }$, and plumes sufficiently separated to remain independent. Splitting the sources increases total fluid entrainment and reduces the interface height by $n^{-2/5}$. A finite source flux cannot simultaneously be neglected and supply all the replacement air; no steady displacement layer without ambient supply explains this restriction.

At steady state, the displacement-ventilation interface lies where the total plume volume flux equals the imposed ventilation flux. For one triangular-profile wall line plume with kinematic buoyancy flux $B_0$, entrainment coefficient $\alpha$, and ventilation flux $A$ per unit span,

A tall opening between fluids of different densities can support simultaneous inflow below a neutral pressure level and outflow above it. For a symmetric opening with a Boussinesq hydrostatic pressure difference and equal discharge coefficients, the neutral level lies at mid-height.

The neutral pressure level is the height at which indoor and outdoor hydrostatic pressures agree. Flow through an opening reverses direction across this level.

A stratified-room plume-exchange model balances the upward volume transported by a warm plume against downward transport by cold wall plumes. A global buoyancy balance relates the source flux to the buoyancy carried out through the vents.