Methodology for the Evaluation of Natural Ventilation in ... - Cham

X HM X HP(5.16)5.3.2 Kinematic SimilarityFor kinematic similarity, the ratios of the fluid velocities and accelerations must be equal. Thiswill ensure that the streamlines and flow patterns will be similar. This requirement is tiedclosely to the dynamic similarity which necessitates that the ratio of all of the forces that causemotion in the operating fluid be equal between the model and prototype. From similarity theory,a scale model will replicate the kinematic boundary conditions of the prototype if the Prandtl,Reynolds, and Archimedes numbers are identical for both cases.The Reynolds number, Re, is the dimensionless ratio between inertial and viscous forces, used indynamic similarity for evaluating the magnitude of these forces.ULRe (5.17)With a low Reynolds number, flow rate has a strong dependence on the viscosity, whereas with ahigh Reynolds number the viscous forces can often be neglected. To meet similarityrequirements:Re (5.18)MRe P UL UL (5.19) M PWhen air is used and the density of air between the model and prototype does not varysubstantially, equation 5.19 can be simplified to:UL M UL P(5.20)The Prandtl number describes the ratio of molecular momentum to thermal diffusivity. Whenevaluating kinematic similarity in the case of buildings, if air is used as the fluid for the model,the Prandlt number is then matched at 0.71. This assumes that the operating temperatures for thebuilding and model fall between 0°C and 100°C. For a Prandtl number less than one, the thermaldiffusivity or speed of heat propagation is larger than the momentum diffusivity. If a fluid otherthan air is used, the Prandtl number must be evaluated and compared to the prototype value. cPr P(5.21)kFinally, the Archimedes number, or the measure of the relative magnitude of buoyancy forces toinertial forces acting on a fluid, is used to evaluate the motion of fluid due to density differences.For ventilation purposes, it provides the ratio of pressure difference associated with buoyancydriven flow to the pressure difference associated with wind driven flow. Recalling equation 2.7for the pressure difference due to buoyancy, and equation 2.9 for the pressure difference due towind, the ratio of buoyancy force to inertial, or wind force is:PBgHTgHT (5.22)22PUUWFor similar wind pressure differences, Δp w can be simplified to U 2 . The dimensionless ratio thenbecomes the Archimedes number, Ar:87