Potential Flow Theory

Transcription

1 .06 Hydodynamics Reading #4.06 Hydodynamics Pof. A.H. Techet Potential Flow Theoy When a flow is both fictionless and iotational, pleasant things happen. F.M. White, Fluid Mechanics 4th ed. We can teat extenal flows aound bodies as invicid (i.e. fictionless) and iotational (i.e. the fluid paticles ae not otating). This is because the viscous effects ae limited to a thin laye next to the body called the bounday laye. In gaduate classes like.5, you ll lean how to solve fo the invicid flow and then coect this within the bounday laye by consideing viscosity. Fo now, let s just lean how to solve fo the invicid flow. We can define a potential function,!( x, z, t), as a continuous function that satisfies the basic laws of fluid mechanics: consevation of mass and momentum, assuming incompessible, inviscid and iotational flow. Thee is a vecto identity (pove it fo youself!) that states fo any scala, ", By definition, fo iotational flow, Theefoe " # "$ = 0 " # V = 0 V = "# whee! =!( x, y, z, t) is the velocity potential function. Such that the components of velocity in Catesian coodinates, as functions of space and time, ae u "! =, v dx "! "! = and w dy dz = (4.) vesion.0 updated 9// A. Techet

2 .06 Hydodynamics Reading #4 Laplace Equation The velocity must still satisfy the consevation of mass equation. We can substitute in the elationship between potential and velocity and aive at the Laplace Equation, which we will evisit in ou discussion on linea waves.! u +! v +! w = 0! x! y! z (4.) "! "! "! " x " y " z + + = 0 (4.3) LaplaceEquation! " # = 0 Fo you efeence given below is the Laplace equation in diffeent coodinate systems: Catesian, cylindical and spheical. Catesian Coodinates (x, y, z) V = uˆ i + vˆ j + wk ˆ = "# i ˆ + "# ˆ j + "# k ˆ = $# "x "y "z " # = $ # $x + $ # $y + $ # $z = 0 Cylindical Coodinates (, θ, z) x y " y = +,! = tan ( x ) V = u e ˆ + u " e ˆ " + u z e ˆ z = #$ # ˆ " # = $ # $ + $# 4 4 $ 3 $ % $ $# ( ' * & $ ) e + #$ e #" ˆ " + #$ e #z ˆ z = %$ + $ # $+ + $ # $z = 0 vesion.0 updated 9// A. Techet

3 .06 Hydodynamics Reading #4 Spheical Coodinates (, θ, ϕ ) x y z " = + +,! cos x ( ) V = u e ˆ + u " e ˆ " + u # e ˆ # = $% $ ˆ =, o x cos! e + " =,! = tan z ( y ) $% $" ˆ e " + $% sin" $# e ˆ # = &% " # = $ # $ + $# $ % $# ( $ # + ' sin+ * $ 3 sin+ $+ & $+ ) sin + $, = 0 $ % $ $# ( ' * & $ ) Potential Lines Lines of constant! ae called potential lines of the flow. In two dimensions d" = #" #" dx + #x #y dy d" = udx + vdy Since d" = 0 along a potential line, we have dy dx = " u v (4.4) Recall that steamlines ae lines eveywhee tangent to the velocity, dy dx = v, so potential u lines ae pependicula to the steamlines. Fo inviscid and iotational flow is indeed quite pleasant to use potential function,!, to epesent the velocity field, as it educed the poblem fom having thee unknowns (u, v, w) to only one unknown (! ). As a point to note hee, many texts use steam function instead of potential function as it is slightly moe intuitive to conside a line that is eveywhee tangent to the velocity. Steamline function is epesented by!. Lines of constant! ae pependicula to lines of constant!, except at a stagnation point. vesion.0 updated 9// A. Techet

4 .06 Hydodynamics Reading #4 Luckily! and! ae elated mathematically though the velocity components: #! #" u = = # x # y #! #" v = = $ # y # x (4.5) (4.6) Equations (4.5) and (4.6) ae known as the Cauchy-Riemann equations which appea in complex vaiable math (such as 8.075). Benoulli Equation The Benoulli equation is the most widely used equation in fluid mechanics, and assumes fictionless flow with no wok o heat tansfe. Howeve, flow may o may not be iotational. When flow is iotational it educes nicely using the potential function in place of the velocity vecto. The potential function can be substituted into equation 3.3 esulting in the unsteady Benoulli Equation. o { ( ) }! # $ p g z t " + $ " + $ +! $ = # 0 #! {" " t } V p " # gz (4.7) $ = 0. (4.8) #! UnsteadyBenoulli $ " + " V + p + " gz = c( t) # t (4.9) vesion.0 updated 9// A. Techet

5 .06 Hydodynamics Reading #4 Summay Potential Steam Function v Definition V = "! V = "#! Continuity "! = 0 Automatically Satisfied (! " V = 0) Iotationality Automatically Satisfied "#("#! ) = "(" $! )% "! = 0!" V = 0 ( )! In D : w = 0, = 0! z v "! = 0 fo continuity! "! z #! = 0 fo iotationality Cauchy-Riemann Equations fo! and! fom complex analysis: # =! + i", whee! is eal pat and! is the imaginay pat Catesian (x, y) "! "! u = u = " x " y "! v = v " y = # "! " x Pola (, θ) u = "! #! u = " #" #! v = v # " = # "! " Fo iotational flow use:! Fo incompessible flow use:! Fo incompessible and iotational flow use:! and! vesion.0 updated 9// A. Techet

6 .06 Hydodynamics Reading #4 Potential flows Potential functions! (and steam functions,! ) can be defined fo vaious simple flows. These potential functions can also be supeimposed with othe potential functions to ceate moe complex flows. Unifom, Fee Steam Flow (D) V =Uˆ i + 0 ˆ j + 0 ˆ k (4.0) #! #" u = U = = # x # y #! #" v = 0 = = $ # y # x (4.) (4.) We can integate these expessions, ignoing the constant of integation which ultimately does not affect the velocity field, esulting in! and!! = Ux and! = Uy (4.3) Theefoe we see that steamlines ae hoizontal staight lines fo all values of y (tangent eveywhee to the velocity!) and that equipotential lines ae vetical staight lines pependicula to the steamlines (and the velocity!) as anticipated. D Unifom Flow: V = ( U, V,0) ;! = Ux + Vy ;! = Uy " Vx 3D Unifom Flow: V = ( U, V, W ) ;! = Ux + Vy + Wz ; no steam function in 3D vesion.0 updated 9// A. Techet

7 .06 Hydodynamics Reading #4 Line Souce o Sink Conside the z-axis (into the page) as a poous hose with fluid adiating outwads o being dawn in though the poes. Fluid is flowing at a ate Q (positive o outwads fo a souce, negative o inwads fo a sink) fo the entie length of hose, b. Fo simplicity take a unit length into the page (b = ) essentially consideing this as D flow. Pola coodinates come in quite handy hee. The souce is located at the oigin of the coodinate system. Fom the sketch above you can see that thee is no cicumfeential velocity, but only adial velocity. Thus the velocity vecto is V = u ˆ e + u " ˆ e " + u z ˆ e z = u ˆ e + 0 ˆ e " + 0 ˆ e z (4.4) and u = Q " = m = #$ # = #% #& u " = 0 = #$ #" = %#& # (4.5) (4.6) Integating the velocity we can solve fo! and! whee! = mln and! = m" (4.7) Q m =. Note that! satisfies the Laplace equation except at the oigin:! x y exclude it fom the flow. = + = 0, so we conside the oigin a singulaity (mathematically speaking) and The net outwad volume flux can be found by integating in a closed contou aound the oigin of the souce (sink): % %% % C! V # nˆ ds = $ # V ds = u d" = Q S o vesion.0 updated 9// A. Techet

8 .06 Hydodynamics Reading #4 Iotational Votex (Fee Votex) A fee o potential votex is a flow with cicula paths aound a cental point such that the velocity distibution still satisfies the iotational condition (i.e. the fluid paticles do not themselves otate but instead simply move on a cicula path). See figue. Figue : Potential votex with flow in cicula pattens aound the cente. Hee thee is no adial velocity and the individual paticles do not otate about thei own centes. It is easie to conside a cylindical coodinate system than a Catesian coodinate system with velocity vecto V = ( u, u, u ) when discussing point votices in a local efeence! z fame. Fo a D votex, u z = 0. Refeing to figue, it is clea that thee is also no adial velocity. Thus, V = u ˆ e + u " ˆ e " + u z ˆ e z = 0 ˆ e + u " ˆ e " + 0 ˆ e z (4.8) whee u = 0 = "# " = "$ "% (4.9) and u " =? = #$ #" = %#& #. (4.0) Let us deive u ". Since the flow is consideed iotational, all components of the voticity vecto must be zeo. The voticity in cylindical coodinates is whee # " uz " u! $ # " u " uz $ # " u! " u $ %& V = ( '! z 0! z ) e + ( ' ) e + ' = " " " z " ( " "! ) e, (4.) * + * + * + vesion.0 updated 9// A. Techet

9 .06 Hydodynamics Reading #4 and u = u + u, (4.) x y u! = u x cos!, (4.3) u = 0 (fo D flow). z Since the votex is D, the z-component of velocity and all deivatives with espect to z ae zeo. Thus to satisfy iotationality fo a D potential votex we ae only left with the z-component of voticity ( e z ) " u! " u # = 0 " "! (4.4) Since the votex is axially symmetic all deivatives with espect θ must be zeo. Thus, "(u # ) " = "u "# = 0 (4.5) Fom this equation it follows that u! must be a constant and the velocity distibution fo a potential votex is K u =!, u 0 =, u z = 0 (4.6) By convention we set the constant equal to! ", whee Γ is the ciculation,. Theefoe u! = # " (4.7) U θ Figue 3: Plot of velocity as a function of adius fom the votex cente. At the coe of the potential votex the velocity blows up to infinity and is thus consideed a singulaity. vesion.0 updated 9// A. Techet

10 .06 Hydodynamics Reading #4 You will notice (see figue 3) that the velocity at the cente of the votex goes to infinity (as! 0 ) indicating that the potential votex coe epesents a singulaity point. This is not tue in a eal, o viscous, fluid. Viscosity pevents the fluid velocity fom becoming infinite at the votex coe and causes the coe otate as a solid body. The flow in this coe egion is no longe consideed iotational. Outside of the viscous coe potential flow can be consideed acceptable. Integating the velocity we can solve fo! and!! = K" and! = " K ln (4.8) whee K is the stength of the votex. By convention we conside a votex in tems of its ciculation,!, whee " =! K is positive in the clockwise diection and epesents the stength of the votex, such that $ #! = " and! = $ ln. (4.9) # " Note that, using the potential o steam function, we can confim that the velocity field esulting fom these functions has no adial component and only a cicumfeential velocity component. The ciculation can be found mathematically as the line integal of the tangential component of velocity taken about a closed cuve, C, in the flow field. The equation fo ciculation is expessed as! = # V " d s C whee the integal is taken in a counteclockwise diection about the contou, C, and ds is a diffeential length along the contou. No singulaities can lie diectly on the contou. The oigin (cente) of the potential votex is consideed as a singulaity point in the flow since the velocity goes to infinity at this point. If the contou encicles the potential votex oigin, the ciculation will be non-zeo. If the contou does not encicle any singulaities, howeve, the ciculation will be zeo. vesion.0 updated 9// A. Techet

11 .06 Hydodynamics Reading #4 To detemine the velocity at some point P away fom a point votex (figue 4), we need to fist know the velocity field due to the individual votex, in the efeence fame of the votex. Equation Eo! Refeence souce not found. can be used to detemine the tangential velocity at some distance o fom the votex. It was given up font that u = 0 eveywhee. Since the velocity at some distance o fom the body is constant on a cicle, centeed on the votex oigin, the angle θ o is not cucial fo detemining the magnitude of the tangential velocity. It is necessay, howeve, to know θ o in ode to esolve the diection of the velocity vecto at point P. The velocity vecto can then be tansfomed into Catesian coodinates at point P using equations (4.) and (4.3). Figue 4: Velocity vecto at point P due to a potential votex, with stength Γ, located some distance o away. vesion.0 updated 9// A. Techet

12 .06 Hydodynamics Reading #4 Linea Supeposition All thee of the simple potential functions, pesented above, satisfy the Laplace equation. Since Laplace equation is a linea equation we ae able to supeimpose two potential functions togethe to descibe a complex flow field. Laplace s equation is "! "! "! " x " y " z #! = + + = 0. (4.30) Let! =! +! whee "! = 0 and "! = 0. Laplace s equation fo the total potential is (!! ) (!! ) (!! ) " + " + " + #! = + + " x " y " z. (4.3) " $! $! # " $! $! # " $! $! # %! = & + ' + & + ' + & + ' ( $ x $ x ) ( $ y $ y ) ( $ z $ z ) (4.3) " $! $! $! # " $! $! $! # %! = & + + ' + & + + ' ( $ x $ y $ z ) ( $ x $ y $ z ) (4.33) " #! = #! + #! = + = (4.34) Theefoe the combined potential also satisfies continuity (Laplace s Equation)! Example: Combined souce and sink Take a souce, stength +m, located at (x,y) = (-a,0) and a sink, stength m, located at (x,y) = (+a,0). ( ) ) ln ( ) ( )! =! souce +! sink = ln m " x + a + y $ x $ a + % y # & This is pesented in catesian coodinates fo simplicity. Recall ( ) mln = mln! x + y " = mln( x + y # ) % $ &! = mln ( ) ( ) x + a + y x " a + y = x + y so that (4.35) (4.36) (4.37) This is analogous to the electo-potential pattens of a magnet with poles at ( ± a,0). vesion.0 updated 9// A. Techet

13 .06 Hydodynamics Reading #4 Example: Multiple Point Votices Since we ae able to epesent a votex with a simple potential velocity function, we can eadily investigate the effect of multiple votices in close poximity to each othe. This can be done simply by a linea supeposition of potential functions. Take fo example two votices, with ciculation! and!, placed at ± a along the x-axis (see figue 5). The velocity at point P can be found as the vecto sum of the two velocity components V and V, coesponding to the velocity geneated independently at point P by votex and votex, espectively. Figue 5: Fomulation of the combined velocity field fom two votices in close poximity to each othe. Votex is located at point (x, y) = (-a, 0) and votex at point (x, y) = (+a, 0). The total velocity potential function is simply a sum of the potentials fo the two individual votices $ $! =! +! =! +! = " + " # # T v v with! and! taken as shown in figue 5. One votex in close poximity to anothe votex tends to induce a velocity on its neighbo, causing the fee votex to move. vesion.0 updated 9// A. Techet