Collision Efficiencies of Ice Crystals at LowâIntermediate Reynolds ...

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1002 JOURNAL OF THE ATMOSPHERIC SCIENCESVOLUME 57inviscid flow fields past disks. Pitter and Pruppacher(1974) and Martin et al. (1981) performed calculationsof the collision efficiency between ice plates and supercooleddroplets assuming that the flow fields pasthexagonal plates can be approximated by that past thinoblate spheroids. Schlamp et al. (1975) calculated theefficiencies with which columnar ice crystals collidewith supercooled droplets assuming that the flow fieldspast an ice column can be approximated by that past aninfinitely long cylinder. While these studies contributedsignificantly to our early understanding of the onset ofriming process, there is room for improvement. Furthermore,all of these studies assumed that flow fieldsare steady, which is not valid for larger ice crystals thatfall in an unsteady attitude (Pruppacher and Klett 1997).Recently, Ji and Wang (1989, 1991) and Wang andJi (1997) performed calculations of the flow fields pastthree different shapes of ice crystals: hexagonal iceplates, broad-branch crystals, and ice columns. Theshapes of ice crystals used in their calculations are morerealistic than those mentioned before. Also, unsteadyfeatures such as eddy shedding were included in thecalculations. These improvements ultimately led tomore accurate computation of flow fields. The presentstudy is based on the flow fields as determined by Wangand Ji (1997). Using these fields we calculated the collisionefficiencies with which ice crystals of the abovethree shapes collided with supercooled droplets. Thedetails of the formulations, results, and conclusions arereported below.2. Physics and mathematicsThe theoretical problem of determining the collisionefficiency between an ice crystal and a supercooledcloud droplet mainly involves the solution of the equationof motion of the droplet in the vicinity of the fallingice crystal. Since the motions occur in a viscous medium,air, the effect of flow fields must be considered.The flow fields around falling ice crystals are complicatedand are normally obtained by solving relevantNavier–Stokes equations governing the flow. The informationof these flow fields is fed into the equationof motion and the latter is solved (usually by numericaltechniques) to determine the ‘‘critical trajectory,’’ thatis, the trajectory of the droplet that makes grazing collisionwith the crystal [see, e.g., chapter 14 of Pruppacherand Klett (1997) for an explanation of the grazingtrajectory]. Finally, the collision efficiency is calculatedbased on the knowledge of the grazing trajectory. In thefollowing, these steps are described one by one.a. Flow fields around falling ice crystalsAs indicated above, the first step of determining thecollision efficiency is to determine the flow fields aroundfalling ice crystals. This is done by solving the incompressibleNavier–Stokes equations for flow past icecrystals:u P2 (u · )u u, (1)twhere u is local flow velocity vector, P the dynamicpressure associated with the flow field, air density and the kinematic viscosity of air. In the context of numericalcalculations, this equation is often nondimensionalizedby utilizing the following nondimensionalvariables:x u tu x , u , t ,a1 u a1P2u aP , Re 12, (2)uwhere x (or y, z) is one of three Cartesian coordinates;a 1 the characteristic dimension of the ice particle; u the free-stream velocity, which is equal to the terminalfall velocity of the ice crystal; and Re is the Reynoldsnumber relevant to the flow. All primed quantities arenondimensional. Using these dimensionless variables,we can write the nondimensional Navier–Stokes equationand the continuity equation as (after dropping theprimes)u 22 u · u P utRe(3) · u 0. (4)The ideal boundary conditions appropriate for thepresent problems areu 0 at the surface of the ice crystal, and (5)u 1·ezat infinity, (6)where e z is a unit vector in the free stream direction.The details of the numerical procedure have been givenin a recent paper by Wang and Ji (1997), so they willnot be repeated here. The velocity vectors so obtainedare input into the equation of motion to be describedbelow.b. Equation of motion and droplet trajectoryThe equation of motion of a cloud droplet of radiusa 2 in the vicinity of a falling ice crystal of characteristicdimension a 1 is2dV d rm m Fg F D, (7)dt dt 2where m is the mass of the droplet, V its velocity, r itsposition vector, F g the buoyancy-adjusted gravitationalforce, and F D the hydrodynamic drag force due to theflow. These two forces are expressed as
15 APRIL 2000 WANG AND JI1003w aF mg , (8)gwhere w and a are the density of water and air, respectively,andCDReFD 6a 2(V u). (9)24In order to calculate the drag force (9), we need to inputthe local flow velocity u at each time step. That valuecomes from the solution of (3).In order to solve Eq. (7), it is also necessary to specifyan initial condition. The appropriate initial conditionhere is the initial horizontal offset y of the droplet fromthe vertical line passing through the center of the fallingice crystal (see Pruppacher and Klett 1997, p. 569).Needless to say, this offset has to be set at a distancesufficiently upstream to ensure that the droplet is progressingin a straight line at the time. In this study theinitial offset was set at 20 radii upstream of the icecrystal and it was proven adequate for the above-statedpurpose. With this initial condition in place, Eq. (7) canbe solved for V and hence r of the droplet as a functionof time. The latter defines its trajectory.To determine the collision efficiency, we need to determinethe critical initial offset y c of the droplet suchthat it will make a grazing collision with the ice crystal.An initial offset greater than y c would result in a miss,whereas one smaller than y c would result in a hit. Inthe present study, a bisection technique, similar to thatused in Miller and Wang (1989), was used to determiney c . Once this is done, the next step is determining thecollision efficiency E. wc. Collision efficiencySince the collector is an ice crystal that is usually nota sphere, it is important to take a closer examination ofthe proper definition of collection efficiency here. Theold definition of collision efficiency based on sphericalsymmetry [e.g., Pruppacher and Klett 1997, Eq. (14-1)]is inappropriate here. The proper definition of E here isthe one given by Wang (1983), namely,K AE , (10)K* A*where K is the collision kernel, K* the geometrical collisionkernel, A the collision cross section, and A* thegeometrical collision cross section (see Fig. 1 for aclearer definition). The relation between the collisionkernel and collision cross section isK A(V u). (11)This definition takes care of the nonspherical shape ofthe ice crystal and is more general than the usual definitionof collision efficiency based on the ‘‘radius’’ ofthe collector that is strictly valid for only spheres. ThusFIG. 1. A schematic explanation of the collision cross section, usinga columnar ice crystal as an example. The hexagon-filled area representsthe collision cross section A, the middle rectangle the crosssection (horizontal projection) of the ice column. The outermost rectanglerepresents the geometrical collision cross section A*.cloud droplets located within A (or K) will be eventuallycollected by the ice crystal and turn into rime. Note thatthe definition of collision efficiency as given by (10)assumes that the droplets are of the same size. The efficiencywould vary if we deal with a distribution ofdrop sizes, but this is not what we treat here.In the case of unsteady flow fields, the trajectory ofa droplet starting from a certain initial offset was determinedby averaging a few trajectories over a cycleof eddy shedding periods. This was done for a few cases,but it was later found to be unnecessary because thesetrajectories vary very little as grazing collisions in thisstudy all occur in the upstream regions where flow fieldsare steady. In addition, droplets are massive enough todefy small fluctuations in the flow fields. This may notbe the case if the collection of submicron particles byhydrometeors is considered since rear capture may occurin that case (e.g., Wang et al. 1978; Wang and Jaroszczyk1991) and the unsteady fields in the downstreamwould have greater effect.3. Results and discussionIce crystal collectors of three different habits are consideredin this study: columnar ice crystals (approximatedas finite circular cylinders), hexagonal ice plates,and broad-branch ice crystals. Tables 1, 2, and 3 showthe dimensions of these ice crystals. These are the sameset of ice crystals whose flow fields and ventilation coefficientswere computed by Wang and Ji (1997) Ji andWang (1999), respectively. Figure 2 shows several trajectoriesof a droplet of 2-m radius moving around afalling broad-branch crystal of Re 10. Of the eighttrajectories shown here, trajectories 1, 2, 6, 7, and 8 aremisses, whereas trajectories 3, 4, and 5 are hits. Trajectory4 is the central trajectory, while 3 and 5 aregrazing trajectories.Note that since the collector ice crystals are notspheres, the critical initial offset y c does not possess
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