LECTURE NOTES - VI. Prof. Dr. Atıl BULU

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1 LECTURE NOTES - VI «FLUID MECHANICS» Istanbl Technical Uniersit College of Ciil Engineering Ciil Engineering Deartment Hdralics Diision

2 CHAPTER 6 TWO-DIMENSIONAL IDEAL FLOW 6. INTRODUCTION An ideal flid is rel hothetical flid, hich is assmed to hae no iscosit and no comressibilit, also, in the case of liqids, no srface tension and aoriation. The std of flo of sch a flid stems from the eighteenth centr hdrodnamics deeloed b mathematicians, ho, b making the aboe assmtions regarding the flid, aimed at establishing mathematical models for flid flos. Althogh the assmtions of ideal flo aear to be far obtained, the introdction of the bondar laer concet b Prandtl in 904 enabled the distinction to be made beteen to regimes of flo: that adjacent to the solid bondar, in hich iscosit effects are redominant and, therefore, the ideal flo treatment old be erroneos, and that otside the bondar laer, in hich iscosit has negligible effect so that idealied flo conditions ma be alied. The ideal flo theor ma also be etended to sitations in hich flid iscosit is er small and elocities are high, since the corresond to er high ales of Renolds nmber, at hich flos are indeendent of iscosit. Ths, it is ossible to see ideal flo as that corresonding to an infinitel large Renolds nmber and ero iscosit. 6.. CONTINUITY EQUATION The control olme ABCDEFGH in Fig. 6. is taken in the form of a small rism ith sides d, d and d in the, and directions, resectiel. B G A C H d D E d d Fig. 6. The mean ales of the comonent elocities in these directions are,, and. Considering flo in the direction, Mass inflo throgh ABCD in nit time ρdd 00

3 In the general case, both secific mass ρ and elocit ill change in the direction and so, ( ρ) Mass otflo throgh EFGH in nit time ρ d dd Ths, ( ρ) Net otflo in nit time in direction ddd Similarl, ( ρ) Net otflo in nit time in direction ddd Therefore, Net otflo in nit time in direction Total net otflo in nit time ( ρ) ddd ( ρ) ( ρ) ( ρ) ddd Also, since ρ/t is the change in secific mass er nit time, Change of mass in control olme in nit time ρ ddd t (the negatie sign indicating that a net otflo has been assmed). Then, Total net otflo in nit time Change of mass in control olme in nit time or ( ρ) ( ρ) ( ρ) ( ρ) ( ρ) ( ρ) ρ ddd ddd t ρ t (6.) Eq. (6.) holds for eer oint in a flid flo hether stead or nstead, comressible or incomressible. Hoeer, for incomressible flo, the secific mass ρ is constant and the eqation simlifies to 0 (6.) For to-dimensional incomressible flo this ill simlif still frther to 0 (6.3) 0

4 EXAMPLE 6.: The elocit distribtion for the flo of an incomressible flid is gien b 3-, 4, -. Sho that this satisfies the reqirements of the continit eqation. SOLUTION: For three-dimensional flo of an incomressible flid, the continit eqation simlifies to Eq. (6.);,, and, hence, 0 Which satisfies the reqirement for continit EULER S EQUATIONS Eler s eqations for a ertical to-dimensional flo field ma be deried b aling Neton s second la to a basic differential sstem of flid of dimension d b d (Fig. 6.). g a d C df D dw A B d df Fig. 6. a The forces df and df on sch an elemental sstem are, df df dd dd ρgdd The accelerations of the sstem hae been deried for stead flo (Eq. 3.5) as, 0

5 a a Aling Neton s second la b eqating the differential forces to the rodcts of the mass of the sstem and resectie accelerations gies, dd gdd dd dd dd ρ ρ ρ and b cancellation of dd and slight arrangement, the Eler eqations of to-dimensional flo in a ertical lane are ρ (6.4) g ρ (6.5) Accomanied b the eqation continit, 0 (6.3) The Eler eqations form a set of three simltaneos artial differential eqations that are basic to the soltion of to-dimensional flo field roblems; comlete soltion of these eqations ields, and as fnctions of and, alloing rediction of ressre and elocit at an oint in the flo field BERNOULLI S EQUATION Bernolli s eqation ma be deried b integrating the Eler eqations for a constant secific eight flo. Mltiling Eq. (6.4) b d and Eq. (6.5) b d and integrating from to on a streamline gie d g d d d d d d ρ ρ 03

6 V d d ds V Hoeer, along a streamline in an stead flo d/d/ and therefore d d. If e collect the both eqations, d g d d d d ρ Since ( ), arranging the eqation ields, ( ) ( ) ( ) ( ) d g d d d d d d ρ Since the terms in each bracket is a total differential, b integrating gies ( ) ( g ρ ) B remembering that V, the eqation takes the form of g V g V γ γ (6.6) This eqation is the ell-knon Bernolli eqation and alid on the streamline beteen oints and in a flo field ROTATIONAL AND IRROTATIONAL FLOW Considerations of ideal flo lead to et another flo classification, namel the distinction beteen rotational and irrotational flo. 04

7 d d ( a ) dθ ( b ) ( a ) Fig. 6. and 6.3 ( b ) Basicall, there are to tes of motion: translation and rotation. The to ma eist indeendentl or simltaneosl, in hich case the ma be considered as one serimosed on the other. If a solid bod is reresented b sqare, then re translation or re rotation ma be reresented as shon in Fig. 6. (a) and (b), resectiel. If e no consider the sqare to reresent a flid element, it ma be sbjected to deformation. This can be either linear or anglar, as shon in Fig. 6.3 (a) and (b), resectiel. The rotational moement can be secified in mathematical terms. Fig.6.4 shos the rotation of a rectanglar flid element in a to-dimensional flo. C' t D' θ A t θ B Fig. 6.4 B' C Dring the time interal t the element ABCD has moed relatie to A to a ne osition, hich is indicated b the dotted lines. The anglar elocit ( AB ) of line AB is, 05

8 ( ) t t t t t AB lim lim 0 0 θ Similarl, the anglar elocit ( AD ) of line AD is t t AD 0 lim θ The aerage of the anglar elocities of these to line elements is defined as the rotation of the flid element ABCD. Therefore, ( ) AD AB (6.7) The condition of irrotationalit for a to-dimensional flo is satisfied hen the rotation is eerhere ero, so that 0 or (6.8) For a three-dimensional flo, the condition of irrotationalit reqires that the rotation abot each of three aes, hich are arallel to, and -aes mst be ero. Therefore, the folloing three eqations mst be satisfied:,, (6.9) EXAMPLE 6.: The elocit comonents in a to-dimensional elocit field for an incomressible flid are eressed as Sho that these fnctions reresent a ossible case of an irrotational flo. SOLUTION: The fnctions gien satisf the continit eqation (Eq. 6.3), for their artial deriaties are and so that 0 06

9 Therefore the reresent a ossible case of flid flo. The rotation of an flid element in the flo field is, [( ) ( )] CIRCULATION AND VORTICITY Consider a flid element ABCD in rotational motion. Let the elocit comonents along the sides of the element be as shon in Fig B d C d Direction of intergration d A d D Fig. 6.5 Since the element is rotating, being art of rotational flo, there mst be a resltant eriheral elocit. Hoeer, since the center of rotation is not knon, it is more conenient to relate rotation to the sm of rodcts of elocit and distance arond the contor of the element. Sch a sm is the line integral of elocit arond the element and it is called circlation, denoted b Γ. Ths, r r Γ V ds (6.0) Circlation is, b conention, regarded as ositie for anticlockise direction of integration. Ths, for the element ABCD, from side AD 07

10 Γ ABCD d d d d d d dd dd Since dd ζ For the to-dimensional flo in the - Ë awe, is the orticit of the element abot the -ais, ζ. ThÔ rodct dd is the area of the element da. Th)s Γ ABCD dd ζ da It is seen, therefore, that the circlation[xrwnd X contor is eqal to the sm of the orticities ithin%the area of contor. This is knon as Stokes thxoem and ma be stated mathematicall, for a general case of an contoé C(Fig_6.6) as Γ C V ds ζ da (6.) s A ζda θ C Fig. 6.6 The aboe considerations<indicate that, for irrotational flo, since orticit iï eqal to ero, Óhe circlation arond a closed contor throgh hich flid és moing, mst be eqal to ero STREAM FUNCTIO A stream fnction ψ is a mathematical deice, hich describes the formœof an articlar attern of flo. In Fig. 6.7 let P (, ) reresent a moable oint in the lane of 08

11 motion of a stead, t o-dimensional flo, and cönsider the flo to hae nit thicrness erendiclar to the -lane. P` q υ P(,) P'' ν Fig. 6.7 The olme rate of flo across an line connecting OP is a fnction of the osition of P and defined as the stream fnction ψ: ψ ( ) f, Stream fnction ψ has a nit of cbic meter er second er meter thickness (normal to the -lane). The to comonents of elocit, and can be eressed in terms of ψ. If the oint P in Fig. 6.7 is dislaced an infinitesimal distance is ψ. Therefore, ψ (6.) Similarl, ψ (6.3) When these ales of and are sbstitted into Eqs. (3.6), the differential eqation for streamlines in to-dimensional flo becomes ψ ψ d d 0 B definition, the left-hand side of this eqation is eqal to the total differential dψ hen ψ f (, ). Ths, and d ψ 0 ψ C (constant along a streamline) (6.4) 09

12 Eq. (6.4) indicates that the general eqation for the streamlines in a flo attern is obtained hen ψ is eqated to a constant. Different nmerical ales of the constant in trn define streamlines. As an eamle, the stream fnction for a stead to-dimensional flo at 90 0 corner (shon in Fig. 6.8) takes the folloing form: ψ The general eqation for the streamlines of sch a flo is obtained hen ψ C (constant), that is, C Which indicates that the streamlines are a famil of rectanglar herbolas. Different nmerical ales of C define different streamlines as shon in Fig Obiosl, the olme rate of flo beteen an to streamlines is eqal to the difference in nmerical ales of their constants C 4 0 υ ν P(,) Fig. 6.8 EXAMPLE 6.3: A stream fnction is gien b ψ 3 3 Determine the magnitde of elocit comonents at the oint (3, ). SOLUTION: The and comonents of elocit are gien b -comonent: ψ ( ) 0

13 ψ comonent: ( ) At the oint (3,) -3 and -8 and the total elocit is the ector sm of the to comonents. r r r V 3i 8 j Note that /0 and /0, so that 0 Therefore the gien stream fnction satisfies the continit eqation. The eqation for orticit, ζ (6.4) ma also be eressed in terms of ψ b sbstitting Eqs. (6.) and (6.3) into Eq. (6.4) ψ ψ ζ Hoeer, for irrotational flos, ζ 0, and the classic Lalace eqation, ψ ψ ψ 0 reslts. This means that the stream fnctions of all irrotational flos mst satisf the Lalace eqation and that sch flos ma be identified in this manner; conersel, flos hose ψ does not satisf the Lalace eqation are rotational ones. Since both rotational and irrotational flo fields are hsicall ossible, the satisfaction of the Lalace eqation is no criterion of the hsical eistence of a flo field. EXAMPLE 6.4: A flo field is described b the eqation ψ -. Sketch the streamlines ψ 0, ψ, and ψ. Derie an eression for the elocit V at an oint in the flo field. Calclate the orticit.

14 0 SOLUTION: From the eqation for ψ, the flo field is a famil of arabolas smmetrical abot the -ais ith the streamline ψ 0 assing throgh the origin of coordinates. ψ ψ ( ) ( ) Which allos the directional arros to be laced on streamlines as shon. The magnitde V of the elocit ma be calclated from 0 V 4 and the orticit b Eq. (6.4) ζ ( ) () sec (Conter clockise) Since ζ 0, this flo field is seen to be rotational one VELOCITY POTENTIAL FUNCTIONS When the flo is irrotational, a mathematical fnction called the elocit otential fnction φ ma also be fond to eist. A elocit otential fnction φ for a stead, irrotational flo in the -lane is defined as a fnction of and, sch that the artial deriatie φ ith resect to dislacement in an chosen direction is eqal to the elocit in that direction. Therefore, for the and directions, φ (6.5)

15 φ (6.6) These eqations indicate that the elocit otential increases in the direction of flo. When the elocit otential fnction φ is eqated to a series of constants, eqations for a famil of eqiotential lines are the reslt. The continit eqation 0 (6.3) ma be ritten in terms of φ b sbstittion Eqs. (6.5) and (6.6) into the Eq. (6.3), to ield The Lalacian differential eqation, φ φ φ 0 (6.7) Ths all ractical flos (hich mst conform to the continit rincile) mst satisf the Lalacian eqation in terms of φ. Similarl, the eqation of orticit, ζ (6.4) ma be t in terms of φ to gie ζ φ φ φ φ from hich a alable conclsion ma be dran: Since, φ φ the orticit mst be ero for the eistence of a elocit otential. From this it ma be dedced that onl irrotational (ζ 0) flo fields can be characteried b a elocit otential φ; for this reason irrotational flos are also knon as otential flos. RELATION BETWEEN STREAM FUNCTION AND VELOCITY POTENTIAL 3

16 A geometric relationshi beteen streamlines and eqiotential lines ma be deried from the foregoing eqations and restatement of certain mathematical definitions; the latter are (ith definitions of and inserted) ψ ψ dψ d d d d φ φ dφ d d d d Hoeer, along a streamline ψ is constant and dψ 0, so along a streamline, d d also along an eqiotential line φ is constant and dφ 0, so along an eqiotential line; d d The geometric significance of this is seen in Fig The eqiotential lines are normal to the streamlines. Ths the streamlines and eqiotential lines (for otential flos) form a net, called a flo net, of mtall erendiclar families of lines, a fact of great significance for the std of flo fields here formal mathematical eressions of φ and ψ are nobtainable. Another featre of the elocit otential is that the ale of φ dros along the direction of the flo, that is, φ 3 <φ <φ. Φ Φ 3 Φ V Const. Streamline 90 υ υ ν ν Fig. 6.9 It is imortant to note that the stream fnctions are not restricted to irrotational (otential) flos, hereas the elocit otential fnction eists onl hen the flo is irrotational becase the elocit otential fnction alas satisfies the condition of irrotationalit (Eq. 6.8). The artial deriatie of in Eq. (6.5) is alas eqal to the artial deriatie in Eq. (6.6) 4

17 For an flo attern the elocit otential fnction φ is related to the stream fnction ψ b the means of the to elocit comonents, and, at an oint (, ) in the Cartesian coordinate sstem in the form of the to folloing eqations: φ ψ (6.8) φ ψ (6.9) EXAMPLE 6.5: A stream fnction in a to-dimensional flo is ψ. Sho that the flo is irrotational (otential) and determine the corresonding elocit otential fnction φ. is, SOLUTION: The gien stream fnction satisfies the condition of irrotationalit, that ψ ψ ( ) ( ) 0 hich shos that the flo is irrotational. Therefore, a elocit otential fnction φ ill eist for this flo. B sing Eq. (6.8) φ ψ ( ) Therefore, φ f( ) (a) From Eq. (6.9) φ ψ From this eqation, ( ) ( ) φ f (b) The elocit otential fnction, φ C 5

18 satisfies both φ fnctions in Eqations a and b. EXAMPLE 6.6: In a to-dimensional, incomressible flo the flid elocit comonents are gien b: 4 and Sho that the flo satisfies the continit eqation and obtain the eression for the stream fnction. If the flo is otential (irrotational) obtain also the eression for the elocit otential. SOLUTION: For incomressible, to-dimensional flo, the continit eqation is 0 bt 4 and - 4. Therefore, 0 and the flo satisfies the continit eqation. To obtain the stream fnction, sing Eqs. (6.) and (6.3) ψ 4 (a) ψ 4 (b) Therefore, from (a), ψ ( 4) f ( ) C f ( ) C Bt, if ψ 0 0 at 0 and 0, hich means that the reference streamline asses throgh the origin, then C 0 and ( ) ψ f (c) To determine f (), differentiate artiall the aboe eression ith resect to and eqate to, eqation (b): ψ f ( ) 4 f ( ) 4 Sbstitte into (c) 6

19 ψ To check hether the flo is otential, there are to ossible aroaches: (a) Since bt 0 ( 4 ) and 4 Therefore, 4 and 4 so that and flo is otential. (a) Lalace s eqation mst be satisfied, ψ ψ ψ 0 ψ Therefore, ψ 4 and ψ 4 ψ 4 ψ and 4 Therefore and flo is otential. No, to obtain the elocit otential, φ 4 7

20 ( ) f ( ) G φ 4 Bt φ 0 0 at 0 and 0, so that G 0. Therefore, φ 4 f ( ) Differentiating ith resect to and eqating to, φ 4 d d f ( ) 4 d d f ( ) and f ( ) so that φ THE FLOW NET In an to-dimensional stead flo roblem, the mathematical soltion is to determine the elocit field of flo eressed b the folloing to elocit comonents: f f (, ) (, ) Hoeer, if the flo is irrotational, the roblem can also be soled grahicall b means of a flo net sch as the one shon in Fig.6.0. This is a netork of mtall erendiclar streamlines and eqiotential lines. The streamlines, hich sho the direction of flo at an oint, are so saced that there is an eqal rate of flo q discharging throgh each stream tbe. The discharge q is eqal to the change in ψ from one streamline to the net. The eqiotential lines are then dran eerhere normal to the streamlines. The sacings of eqiotential lines are selected in sch a a that the change in elocit otential from one eqiotential line to the net is constant. Frthermore, that is, ψ φ. As a reslt the form aroimate sqares (Fig. 6.0) 8

21 Sqare Eqiotential lines q q q Streamlines q Fig.6.0 From the continit relationshi, the distances beteen both sets of lines mst therefore be inersel roortional to the local elocities. Ths the folloing relation is a ke to the roer constrction of an flo net. n n s s Where n and s are resectiel the distance beteen streamlines and beteen eqiotential lines. Since there is onl one ossible attern of flo for a gien set of bondar conditions, a flo net, if roerl constrcted, reresents a niqe mathematical soltion for a stead, irrotational flo. Wheneer the flo net is sed, the hdrodnamic condition of irrotationalit (Eq. 6.8) mst be satisfied. The flo net mst be sed ith cation. The alidit of the interretation deends on the etent to hich the assmtion of ideal (noniscos) flid is jstified. Fortnatel, sch flids as ater and air hae rather small iscosit so that, nder faorable conditions of flo, the condition of irrotationalit can be aroimatel attained. In ractice, flo nets can be constrcted for both the flo ithin solid bondaries (Fig.6.) and flo arond a solid bod (Fig. 6.). υ 0 Fig. 6. 9

22 υ 0 Fig. 6. In either flo the bondar srfaces also reresent streamlines. Other streamlines are then sketched in b ee. Net, the eqiotential lines are dran eerhere normal to the streamlines. The accrac of the flo net deends on the criterion that both sets of lines mst form aroimate sqares. Usall a fe trials ill be reqired before a satisfactor flo net is rodced. After a correct flo net is obtained, the elocit at an oint in the entire field of flo can be determined b measring the distance beteen the streamlines (or the eqiotential lines), roided the magnitde of elocit at a reference section, sch as the elocit of flo 0 in the straight reach of the channel in Fig. 6., or the elocit of aroach 0 in Fig. 6., is knon. It is seen from both that the magnitde of local elocities deends on the configration of the bondar srface. Both flo nets gie an accrate ictre of elocit distribtion in the entire field of flo, ecet for those regions in the icinit of solid bondaries here the effect of flid iscosit becomes areciable. 6.. GROUND WATER FLOW The flo net and flo field serosition techniqes ma also be alied to the flo of real flids nder some restrictions, hich are freqentl encontered in engineering ractice. Consider the one-dimensional flo of an incomressible real flid in a stream tbe. The Bernolli eqation ritten in differential form is V d γ g dh L Sose no that V is small (so that dv /g ma be neglected) and the head loss dh L gien b dh L Vdl (6.0) K 0

23 in hich dl is the differential length along the stream tbe and K is a constant. The Bernolli eqation aboe then redces to d Vdl γ K V d dl K γ and, if this ma be etended to the to-dimensional case, φ K γ φ K γ (6.) (6.) and K γ is seen to be the elocit otential of sch flo field. The conditions of the foregoing hothetical roblem are satisfied hen flid flos in a laminar condition throgh a homogenos oros medim. The media interest are those haing a set of interconnected ores that ill ass a significant olme of flid, for eamle, sand, and the certain rock formations. The head-loss la (Eq. 6.0) is sall ritten as V dh K dl L K dh dl (here h/γ) and is an eerimental relation called Darc s la; K is knon as the coefficient of ermeabilit, has the dimensions of elocit, and ranges in ale from m/sec for cla to 0.3 m/sec for grael. A Renolds nmber is defined for oros media flo as Re Vd/ν, here V is the aarent elocit or secific discharge (Q/A) and d is a characteristic length of the medim, for eamle, the effectie or median grain sie in sand. When Re< the flo is laminar and Darc s linear la is alid. If Re> it is likel that the flo is trblent, that V /g is not negligible, and Eq. 6.0 is not alid. Note that V is not the actal elocit in the ores, bt is the elocit obtained b measring the discharge Q throgh an area A. The aerage elocit in the ores is V V/n here n is the orosit of the medim; n(volme of oids) / (Volme of solids ls oids) Een thogh the actal flid flo in the oros medim is iscos dominated and rotational, the aarent flo reresented b V and the elocities and (Eqations 6. and 6.) is irrotational. Both the flo net and serosition of flo field concets can be sed. The flo net is er sefl in obtaining engineering information for the seeage flo of ater throgh or nder strctres, to ells and nder drains, or for the flo of etrolem

24 throgh the oros materials of sbsrface reseroirs. Flo field serosition is most sefl in defining the flo attern in grond ater aqifers nder the action of recharge and ithdraals ells.