DEPATMENT OF CIVIL AND ENVIONMENTAL ENGINEEING FLID MECHANICS III Solutions to Poblem Sheet 3 1. An tmospheic vote is moelle s combintion of viscous coe otting s soli boy with ngul velocity Ω n n iottionl vote outsie the coe. Ae continuity n iottionlity conitions stisfie fo this moel? Cn Benoulli eqution be use to clculte pessue genete by the vote? Cn supeposition pinciple be use to escibe intection of such vote with othe objects? As n elementy volume we use n element between coointe lines in cylinicl coointes: < < + ; Θ < θ < Θ + θ. The flow is isymmetic (no chnges in cicumfeentil iection ( )/ θ = 0), which mens tht flows though = 1 n = e the sme. The il velocity component is zeo (v θ = 0) n thee is no flow in the il iection. Theefoe the totl flow though the bouny of ny elementy volume is zeo, n the continuity conition is stisfie. Cicumfeentil velocity component insie the coe is v θ = Ω n outsie the coe v θ = Γ/(π ), whee Ω n Γ e some constnts. The cicultion oun n elementy contou C enclosing the elementy volume is: Γ C = v θ ( + ) ( + ) θ v θ () θ. Insie the coe: Γ C = Ω ( θ ( + ) θ ) = Ω A C /, whee A C is the e of n elementy contou. Outsie the coe: Γ C = Γ ( 1 + ( + δ) 1 ) = 0. The flow is iottionl outsie the coe, but ottionl insie the coe. The Benoulli eqution cn be pplie to the whole egion outsie the coe, bot not insie. The supeposition pinciple cn be pplie outsie the coe if petubtions cete by objects e smll enough not to influence the flow insie the coe.. The velocity fiel genete by fee vote n by D souce e v = 0; v τ = Γ fo the vote n v = ; v τ = 0 fo the souce, whee v n v τ e the il n cicumfeentil components of the velocity vecto () Cn the supeposition pinciple be pplie to these flows? (b) The souce n the vote with intensities = 1 n Γ = 1 e both plce in the cente of the coointe system = 0. On piece of gph ppe plot the vecto fiel of velocity of the combine flow. Wht is the shpe of stemlines? (c) Is the esulting flow continuous n iottionl n why? (,c) Both flows e continuous n iottionl n theefoe line, which mens tht the supeposition pinciple cn be pplie, n the esult of the supeposition will lso be continuous n iottionl. 1
(b) The il velocity component equls to the cicumfeentil component. This mens tht velocity vectos t evey point hve the ngle of 45 with cicles n ys of the il coointe system (Fig. 1). The vlue of the vecto is invesely popotionl to the ius Fig. 1 V = v + v θ = + Γ The coesponing stemlines e Achimees spils. 3. () Wht is velocity istibution n flow ptten fo 3-imensionl souce/sink. Wite own eltions fo velocity components in suitble coointe system. Show tht the continuity conition is stisfie. (b) Descibe min fetues of combintion of unifom flow of velocity V n 3D souce of stength. Wht is the shpe of boy which cn be moelle by such combintion? (c) An iship in unifom flow of velocity V is moelle by couple souce-sink of stength on the is of the iship, which is pllel to the flow. Fin the length of the iship if the istnce between the souce n the sink is. + Fig. Fig. 3 () We use spheicl coointes (, φ, θ) with the souce (sink) t the oigin = 0, whee is the ius, φ is the pol ngle n θ is the zimuthl ngle. The flow hs spheicl symmety n only the il velocity component is not zeo: ( )/ φ = ( )/ θ = 0; V θ = V φ = 0 n V = V (). The flow though ny sphee enclosing the souce equls to the souce flow. Thus, we hve = V () 4 π, which gives V () = 4 π.
To show the continuity chose n elementy volume between coointe sufces n use flow symmety. (b) Both the unifom flow n the 3D souce flow e continuous n iottionl, n in the cse of n iel flui the supeposition pinciple cn be pplie to the combintion of these two flows (lineity). Due to symmety of both flows the combine flow will be isymmetic with the symmety is pllel to the unifom flow velocity (see Fig. ). Thee is point on the is upstem of the souce whee flow velocity is zeo (the citicl point): = V = 1 =. 4 π 4 π The citicl point belongs to sufce of ottion which completely septes flow of the souce fom the unifom flow. This sufce is stem sufce (no flui cosses it) n it cn be use s sufce of semi-infinite boy of ottion in the unifom flow. F ownstem the souce velocity becomes negligible compe to the unifom flow velocity n the velocity of the combine flow ppoches. Theefoe, the boy sufce ppoches cyline pllel to the unifom flow. The ius of this cyline cn be obtine fom the conition tht ll flui fom the souce emins insie the boy: = π =. π (c) The length of the moelle iship is the istnce between upstem n ownstem citicl points L = + (see Fig. 3). The istnce cn be clculte fom the conition of zeo velocity: 4 π + 4 π ( + ) = 0. The esulting eqution is of the 4 th oe n cn not be esily solve. Howeve, if we ssume the iship to be slene boy (length much lge thn the imete) then s n +. Then we hve: n L = +. 4 π + 4 π 0 = + 4 π 4. A oof of hng in unifom win is moelle by spheicl souce n n equl sink plce on the is of the hng t istnce = 50 m between ech othe (Fig.4). The miml ius of the hng is = 10 m. The win blows long the hng is with velocity = 10 m/s. Assuming the length of the hng is much gete thn its ius () Clculte the souce stength equie to moel the hng. (b) Clculte foce pplie to 1 m htch t the top of the hng if the pessue insie equls to miml pessue t its oute sufce. 3
00000000000000000000000000000000000000000000000000000000000000000000000 u q + β 00000000000000000000000000000000000000000000000000000000000000000000000 11111111111111111111111111111111111111111111111111111111111111111111111 11111111111111111111111111111111111111111111111111111111111111111111111 / 0000000000000000000000000000000000 1111111111111111111111111111111111 0000000000000000000000000000000000 1111111111111111111111111111111111 + Fig. 4 () Flui fom the uppe hlf of + stys insie the sufce. ne the ssumption of slene boy the velocity genete by the souce n sink t the cente of the hng is lmost pllel n unifom. The velocity t the centl coss-section is then u = + u q + 4 π (/). The flow te though the centl coss-section ue to this velocity u = u π / shoul be equl /. This gives = π 1 ( / ). (b) Totl velocity t the htch is u = + u q. We hve This gives u q = 4 π cos(β); cos(β) = /; = /4 +. u = + 4 π ( /4 + ) 3/. Applying Benoulli the pessue t the htch is P = ρ ( u ), n the pessue insie the hng is the stgntion pessue P 0 = ρ /. This gives foce pplie to the htch s F = A (P 0 P ). 5. A D souce of stength is plce t istnce bove hoisontl sufce. Wht is iection n vlue of the foce cting on the souce. 4
0 α V V τ V Fig. 5 The flow fom souce stisfies both iottionlity n continuity conitions. This mens tht fo the cse of n iel flui the flow is line n the eflection pinciple cn be pplie (see Fig. 5). The noml velocity component t the hoisontl sufce fom the combintion of the two souces is zeo (the bouny conition is stisfie). The tngentil velocity component V τ cn be clculte s the vecto sum of velocities fom both souces: V τ = cos(α) = = π = π +, whee is the coointe mesue long the plte with = 0 just behin the souce. We cn now pply the Benoulli eqution to clculte pessue s function of the hoisontl coointe : P + ρ V = 0 P = ρ ( + ), whee the tmospheic pessue t = ± is zeo. Without the souce the pessue ove the whole plte is zeo. The souce les to pessue eistibution n the itionl pessue foce is pplie to the sufce ue to the souce pesence. Accoing to the Newton s 3 lw the opposite foce will be pplie to the souce. Thus fo the D foce (foce pe unit with) we hve: F = P () = ρ = ρ ( + ) = ρ 4. 5