The Equations of Motion for Thermally Driven, Buoyant Flows
نویسنده
چکیده
In this paper a set of approximate equations is derived which is applicable to very nonadiabati c, nondissipative, buoyant flows of a perfect gas. The flows are assumed to be generated by a heat source in which the heat is added slowly. The study is motivated by the occurrence of such f] ows in fires. There, the time scale associated with the fire growth and resultant f]uid moti on is usually long compared with the transil lime of an acoustic signal (based on the temperature derived from the heat added) across the spatial extent of the fire. The appro' mate equations are characterized by a spatially uniform mean pressure appearing in both the energy equa vn and the equation of state with the spat iall y nonuniform portion of the pressure on ly appearing in Ihe momentum equation . Therefore, the pressure remains almost constant in space while significant density and te mperature variations, such as might occur in a fire, are allowed. The approximate equations are shown to reduce to Ihe Boussinesq equations when the heat add ition is mild. These equations are also shown in general to admit internal-wave motions while " filtering out" high-frequency, acoustic waves. ]n addition, they are shown to be express ible in conservation form, the pressure sati sfying an elliptic equation whose homogeneous terms are d e rivable from the wave equation by le tting the sound speed become infinite. An equation for the mean pressu re is also obtained. For the special case of a room heated al a unifonn rate with a small leak to the outside, an approximate solution for the mean pressure is determined expli citl y.
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تاریخ انتشار 2010