円板の抵抗と円形の孔を通る流れの抵抗との相似性
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(2) Vol. 20, No. 1 Journal of Hokkaido University of Education (Section II A) September 1969. Similarity between the Drag of a Circular Disk and That of Flow through a Circular Hole. Takashi SAWADA The Department of Physics, Asahigawa Branch, Hokkaido University of Education. R EB ^ ±. w®®^ wo?L^a%^z®gsu ®ffi?fe Summary The definition of the drag coefflcient of a circular disk moving broadside-on in a fluid is well known and solved at low Reynolds' number. On the other hand the relationship between a pressure drop experienced by the flow through a circular hole and a volumetric flow rate, viscosity of the fluid and radius of the hole are also known at low Reynolds' number. The pressure drop is regarded as the drag per unit area of the hole. Hence the pressure drop multiplied by the area of the hole is the total drag of the flow through the hole. Therefore the drag coefficient of a circular hole can be defined as being similar to a circular disk and shown by a simple formula. Such a formulation is only a reformation of the already known relationship, but the drag coefiicient of a hole is perhaps a new concept, for it has been used only to a body up to date and a hole is not body. The concept of the drag coefficient of a hole is useful to understand the drag of the flow through a hole and can also be applied to the flow through a wire net.. The problem of the flow through a circular hole in a thin plane wall is solved by R. Sampson.0 According to his solution the pressure drop experienced by the fluid flowing through the hole is given by. AP=3^. ^. (D. where /\P is the pressure drop; q, the volumetric flow rate; /(, the viscosity of the fluid and •I', the radius of the hole. Denoting the mean velocity of the flow by Vm and the density of the fluid by p, we can express Reynolds' number of the flow as. 7?.=. 2rvmp. /1 Eq (1) is ascertained experimentally when. Re. <6.4. (2). (3). Now we can regard the pressure drop as the drag per unit area of the hole, hence the total drag of the flow through a circular hole is expressed as D=AP.^=3-w2y:;/^""=37r^,,, :g—— = 37T I'/.lVm. (4).
(3) Takashi Sawada On the other hand the drag coefficient, C, of any body is defined as. D. =-|-Cp5^. (5). where S is a projected area of the body on the plane normal to the free stream ; v, the velocity of the free stream. In the case of a circular hole we use the mean velocity of the flow through a hole instead of the velocity of the free stream. Hence Eq (5) should be rewritten as D=-J-C'p5y2,n. (6). Then, by combining Eq (4) and (6) we get C'=-^T/L rvmp. (7). Inserting Eq (2) into Eq (7) the following relation is obtained : ^_ 127T. 7=^. (8). The drag and the drag coefficient of a circular disk moving broadside-on in a fluid at low Reynolds' number are given by D=16^rv=^-CpTci'W. Re=2^. c=l^ c--^e. (9). (10). <u) <12). Tliese equations are due to M. Ray.2''. Eq (4) shows the same dependency as Eq (9) with regard to the radius, the viscosity and the velocity except for the numerical factor. Hence the drag of a circular hole has essentially the same character as the drag of a circular disk. Therefore we can treat a hole as if were a body thrusting a drag. Assuming the same value of the Reynolds' number we obtain. %^-=^2=1.85, lisk) 64 ^'""'. (13). namely the drag of a circular hole is approximately two times of the drag of a circular disk. Eq (1) has the same physical contents with Eq (4), but it has some different physical meanings.. The concept of the drag of a hole is especially useful to treat the drag of a wire net. The drag of a wire net can be regarded as the drag of a lattice of many circular cylinders, but also as the drag of the square of rectangular holes. The author has been working u;;perimentally the drag of wire net from such a point of view as described above.. References 1) R. Sampson, Phil. Trans. Roy. Soc. A 182 (1891), 449.. 2) M. Ray, Phil. Mag. (Ser. 7) 21 (1936), 546.. (5).
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