Radial Inflow Expeiment:GFD III John Mashall Febuay 6, 003 Abstact We otate a cylinde about its vetical axis: the cylinde has a cicula dain hole in the cente of its bottom. Wate entes at a constant ate though a diffuse on its oute wall and exits though the dain; a steady state is set up in which the flow down the cental dain exactly balances the inflow fom the oute edge. The wate flows inwads fom the diffuse conseving angula momentum and, in so doing, acquies a swiling motion which exhibits a numbe of impotant pinciples of otating fluid dynamics - consevation of angula momentum, geostophic (and cyclostophic) balance. 1 Intoduction We ae all familia with the swil and gugling sound of wate flowing down a dain. Hee we set up a laboatoy illustation of this phenomenon and study it in otating and non-otating conditions. We otate a cylinde about its vetical axis: the cylinde has a cicula dain hole in the cente of its bottom - see Fig.1. Wate entes at a constant ate though a diffuse on its oute wall and exits though the dain; a steady state is set up in which the flow down the cental dain exactly balances the inflow fom the oute edge. The wate flows inwads fom the diffuse conseving angula momentum and, in so doing, acquies a swiling motion, as sketched in Fig.. The swiling motion can become vey vigoous if the cylinde is otated even at only modeate speeds, because the angula momentum of the cylinde is concentated by inwad flowing ings of fluid. The swiling flow exhibits a numbe of impotant pinciples of otating fluid dynamics - consevation of angula momentum, geostophic (and cyclostophic) balance, all of which will be studied in detail in this chapte and made use of in ou subsequent discussions. The expeiment also gives us an oppotunity to think about fames of efeence. 1
THE APPARATUS AND OBSERVED FLOW PATTERNS Figue 1: Sketch of adial inflow appaatus. A diffuse with a 30 cm inside diamete is constucted of wie sceen (and filled with stones appoximately 1cm in size), is placed in a lage tank. Wate is then fed evenly in to the bottom of the diffuse. The diffuse is effective at poducing an axiallysymmetic, inwad flow at the sceen. Below the tank thee is a lage catch basin, patially filled with wate and containing a submesible pump whose pupose is to etun the wate to the diffuse in the uppe tank. The whole appaatus is then placed on a tuntable. The expeiment descibed hee was designed by Jack Whitehead of the Woods Hole Oceanogaphic Institution. Fo moe details efe to Whitehead, J.A and Potte, D.L (1977) Axisymmetic citical withdawal of a otating fluid. Dynamics of Atmosphees and Oceans,, 1-18. The appaatus and obseved flow pattens We take a cylindical tank with a dain hole in the cente of the bottom - see Fig.1. A diffuse is effective at poducing a symmetical inflow towad the dain. The entie appaatus is mounted on a tuntable and viewed fom the laboatoy fame and, using a camea mounted above co-otating with the appaatus, fom the otating fame. The table is tuned in an anticlockwise diection (in the same diection as the spinning Eath). The path of fluid pacels is tacked by dopping pape dots on the fee suface. When the appaatus is not otating, wate flows adially inwad fom the diffuse to the dain in the middle, as sketched in Fig. (lhs). The fee suface is obseved to be athe flat. When the appaatus is otated, howeve, the wate acquies a swiling motion: fluid pacels spial inwad as sketched in Fig. (hs). Even at modest otation ates of Ω =1adian pe second (coesponding to a otation peiod of aound 6 seconds) 1,theeffect of otation is 1 Note that if Ω is the ate of otation of the tank in adians pe second, then the peiod of otation is
3 EXPERIMENTS AND MEASUREMENTS 3 Figue : Flow pattens (left) in the absence of otation and (ight) when the appaatus is otating in an anticlockwise diection. Figue 3: The fee suface of the adial inflow expeiment viewed in the laboatoy fame, in the case when the appaatus is apidly otating. The cuved suface povides a pessue gadient foce diected inwads that is balanced by an outwad centifugal foce due to the anticlockwise ciculation of the spialing flow. maked and pacels complete many cicuits befoe finally exiting though the dain hole. In the pesence of otation the fee suface becomes makedly cuved, high at the peiphey and plunging downwads towad the hole in the cente, as shown in the photogaph - see the photogaph in Fig.3. 3 Expeiments and measuements The main object of ou expeiment is to measue, and intepet in tems of angula momentum pinciples, the velocity field, v θ (), and its connection to the pessue field given by the heightofthefeesufaceh(). Set the tank otating at a ate of 10 pm (evolutions pe minute), tun on the pump and ecod the flow ate, Q. The task is then to measue the suface flow and fee suface τ tank = π Ω.Thusifτ tank =πs, thenω =1.
4 THEORY 4 pofiles and compae them with pedictions based on the theoy in Section 4. Ty vaious otation ates and flow ates. 1. Velocity. Measue the velocity of paticles (black pape dots) moving with the flow in the fee suface of the fluid. This can be done by ecoding a sequence of images using the ovehead camea and making use of the paticle tacking softwae on the laboatoy computes. This will etun the coodinates of individual paticles as a function of time (fame numbe). Compute both the azimuthal and adial velocity. Check whethe the azimuthal speed of the dots, v θ (), is consistent with angula momentum consevation, Eq.(11) below. Compute the Rossby numbe given by Eq.(8). How does it compae to the theoetical pediction, Eq.(1)?. The height field. Measue the depth, H, of the wate at the adius 1 and estimate H at othe adii. Use you measuements of adial velocity, v, and adial volume flux, Q, toinfeh() using Eq.(13). Aethegadientsofthefeesufacesufficient to balance the centifugal acceleation in Eq.(3)? 4 Theoy 4.1 Dynamical balances In the limit in which the tank is otated apidly, pacels of fluid ciculate aound many times befoe falling out though the dain hole; the pessue gadient foce diected adially inwads (set up by the fee suface tilt) is balanced by a centifugal foce diected adially outwads. If V θ istheazimuthalvelocityintheabsolutefame(thefameofthelaboatoy)andv θ is the azimuthal speed elative to the tank (measued using the camea co-otating with the appaatus) then (see Fig. 4): V θ = v θ + Ω (1) whee Ω is the ate of otation of the tank in adians pe second. Note that Ω is the azimuthal speed of a paticle fixed to the tank at adius fom the axis of otation. We now conside the balance of foces in the vetical and adial diections, expessed fist in tems of the absolute velocity V θ and then in tems of the elative velocity v θ.
4 THEORY 5 Figue 4: The velocity of a fluid pacel viewed in the otating fame of efeence: v ot =(v θ,v ). 4.1.1 Vetical foce balance We suppose that hydostatic balance petains in the vetical: p + ρg =0whee ρ is the z density, g is the acceleation due to gavity and z is a vetical coodinate. Integating in the vetical and supposing that the pessue vanishes at the fee suface (actually p = atmospheic pessue at the suface, which hee can be taken as zeo), we find that (ρ and g ae constant): p = ρg (H z) () whee H() is the height of the fee suface and we suppose that z =0(inceasing upwads) on the base of the tank. 4.1. Radial foce balance in the non-otating fame Ifthepitchofthespialtacedoutbyfluid paticles is tight (i.e. in the limit that v v θ << 1, appopiate when Ω is sufficiently lage ) then the centifugal foce diected adially outwads acting on a paticle of fluid is balanced by the pessue gadient foce diected inwads associated with the tilt of the fee suface. This adial foce balance can be witten in the non-otating fame thus: V θ = 1 p ρ. Using Eq.(), the adial pessue gadient foce in the above can be diectly elated to the gadient of the fee suface enabling the foce balance to be witten: Note that this assumption is elaxed in the appendix.
4 THEORY 6 V θ = g H (3) 4.1.3 Radial foce balance in the otating fame Using Eq.(1), we can expess the centifugal acceleation in Eq.(3) in tems of velocities in the otating fame thus: Hence Vθ = (v θ + Ω) = v θ +Ωv θ + Ω (4) vθ +Ωv θ + Ω = g H ³ Theabovecanbesimplified by witing Ω = h = H Ω Ω and defining a quantity h: (5) g, (6) the height of the fee suface measued elative to that of the efeence paabolic suface Ω (see notes on paabolic suface ). Then Eq.(5) can be witten in tem of h thus: v θ = g h Ωv θ: Gadient wind (7) Eq.(3) and Eq.(7) ae completely equivalent statements of the balance of foces. The distinction between them is that the fome is expessed in tems of V θ,thelatteintems of v θ. NotethatEq.(7)hasthesamefomasEq.(3)exceptanextatem, Ωv θ,appeas on the hs of Eq.(7) - this is called the Coiolis acceleation. It has appeaed because we have chosen to expess ou foce balance in tems of elative, athe than absolute velocities. Let us compae the magnitude of the v θ and Ωv θ tems in Eq.(7). Thei atio is the Rossby numbe : R o = v θ (8) Ω If R o << 1, the v θ tem can be neglected in (7). In this limit, Coiolis and pessue gadient tems balance one anothe. Ωv θ = g h : geostophic balance (9) Equation (9) is a simple fom of the geostophic equation elating velocities in the otating fame to the hoizontal pessue gadient in the limit of small R o. So, how lage g
4 THEORY 7 is R o in ou expeiment? We can estimate its size by computing v θ basedonangula momentum consevation. 4. Angula momentum Fluid enteing the tank at the oute wall will have angula momentum because the appaatus is otating. As pacels of fluid flow inwads they will conseve this angula momentum (povided that they ae not ubbing against the bottom o the side). Consevation of angula momentum states that: V θ = constant = Ω 1 (10) Hee 1 is the inne adius of the diffuse in Fig.(1) and V θ is the azimuthal velocity in the laboatoy (inetial) fame given by Eq.(1). Combining Eqs.(10) and (1) we find: v θ = Ω ( 1 ) (11) and hence, fom ou definition of R o, Eq.(8): R o = 1 ³ 1 1. (1) Thus R o =1at = 1 3 ; R o < 1 if > 1 3 (the egion of geostophic balance) and R o > 1 if < 1 3 (the egion of cyclostophic balance - a tonado egion!). We see that in the oute egions of the flow the inwad adial pessue gadient is balanced by outwad Coiolis foces (small R o ): the flow is in geostophic balance hee. But as pacels spial in to the dain they pass though a egion whee R o becomes inceasingly lage and v θ in Eq.(7) becomes a dominant tem. Cyclostophic balance is the one in which the centifugal tem dominates the Coiolis tem in Eq.(7): v θ 4.3 The mass balance = g h : cyclostophic balance If the volume souce coming adially inwads though the diffuse has stength Q, then consevation of volume tells us that: whee V () is the adial velocity at adius. πhv = Q (13)
5 APPENDIX 8 5 Appendix Eq.(3) is an appoximate statement of adial foce balance in the limit that the pitch of the spial taced out by fluid paticles is tight (i.e. in the limit that v v θ << 1). If this is not tue then we must include adial acceleations and use the following moe accuate statement of adial momentum equation: V Vθ V = g H adial acc n centifugal acc n Hee V is the adial component of velocity. pessue gadient (14) 5.0.1 Solutions Using Eqs.(10) and (13), Eq.(14) can be witten: µ Q Ω 1 4 8π H 3 which, on integation, can be witten: = g H Q gh + 8π H + Ω 1 4 = constant (15) whee H 1 is the depth of the wate at adius 1.Eq.(15)isacubicfoH which can be solved. V Use you velocity measuements of estimate how lage is the tem V Eq.(14) in you expeiment. Use (13) to show that if Q πhω1 then the foce balance Eq.(14) educes to Eq.(3). condition that v v θ << 1? elative to V θ << 1 (16) in Can you see that Eq.(16) is just the