Cascade theory. The theory in this lecture comes from: Fluid Mechanics of Turbomachinery by George F. Wislicenus Dover Publications, INC.

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1 Caade theory The theory i thi leture ome from: Fluid Mehai of Turbomahiery by George F. Wilieu Dover Publiatio, INC. 1965

2 d = dt 0 = + Y ρ 0 = p + = kot. F Y d F X X Cotour i the hage of loity due to the vae The otour i large ompared to the

4 Deompoe the veloity i the ormal ad the tagetial diretio of the otour = = ( oα + ) + ( i α + ) ( o α + i α) + v ( oα + i α) + + = + ( oα + i α) +

5 Beroulli equatio ( ) ( ) 0 0 i o p p p p + α α + + ρ + = ρ + =

6 Fore i the x-diretio The fore i the x-diretio atig o the elemet d a be alulated a a fore omig from preure ad impule. df x = p oα ρ ρ ( ) ( ) oα + oα + o ( oα + ) ( i α + ) i α α Flow Rate, Q Veloity i x-diretio, x

7 Fore i the x-diretio We iert the equatio for the preure, p from Beroulli equatio. p = p ρ ( oα + i α) ( + + ) 0 + df x = p oα ρ ρ ( ) ( ) oα + oα + o ( oα + ) ( i α + ) i α α

8 Fore i the x-diretio We iert the equatio for the preure, p from Beroulli equatio. p = p ρ ( oα + i α) ( + + ) 0 + df x = p 0 ρ + ρ ρ oα ( oα + i α) ( + + ) ( ) ( ) oα + oα + o ( oα + ) ( i α + ) i α oα α

9 Fore i the x-diretio df x = p 0 + ρ ρ ρ oα oα + o α + oα i α + oα 3 ( o α + oα + o α) ( oα i α + i α + oα i α + i α)

10 Fore i the x-diretio The hage of veloity, i very mall beaue the large ditae from the airfoil to the otour. We eglet the term that ha the eod order of. df x = p 0 + ρ ρ ρ oα oα + o α + oα i α + oα 3 ( o α + oα + o α) ( oα i α + i α + oα i α + i α)

11 Fore i the x-diretio df x = p 0 oα + ρ 1 oα o α i α ρ o α + i α df x = p 0 oα ρ oα ρ Thi i the fore atig i the x-diretio o a mall elemet, d of the otour.

12 Fore i the x-diretio df x = p0 oα ρ oα ρ By itegratig aroud the otour, we will fid the total fore atig i the x-diretio. =0 =0 F x = p 0 oα ρ oα ρ F x = ρ

13 d Alembert paradox Fx = ρ = 0 The term d i the flow rate through the otour. If the flow i iompreible, the itegral of the term d aroud the otour will be zero. A body i a two-dimeioal ad oviou flow with otat eergy will ot exert a fore i the diretio parallel uditurbed flow,

14 Fore i the y-diretio The fore i the y-diretio atig o the elemet d a be alulated a a fore omig from preure ad impule. df y = p i α ρ ρ ( ) ( ) oα + oα + i α ( oα + ) ( oα + ) oα

15 Fore i the y-diretio df y = p0 i α ρ i α ρ Thi i the fore atig i the y-diretio o a mall elemet, d of the otour.

16 Krefter i y-retig df y = p0 i α ρ i α ρ By itegratig aroud the otour, we will fid the total fore atig i the y-diretio. =0 =0 F y = p 0 i α ρ i α ρ F y = ρ

17 Lift F y = ρ

18 Cirulatio Γ = d

19 Lift F y = ρ Γ

20 The law of the irulatory flow about a defletig body I the abee of ay defletig body iide the hathed area of the otour the fore i y- diretio mut eearily be zero. Thi lead to the theorem that: For a flow of otat eergy the irulatio aroud ay loed otour ot eloig ay fore-tramittig body mut be zero.

21 The law of the irulatory flow about a defletig body Let the irulatio aroud the outer otour i the figure be: Γ 1 = Let the irulatio aroud the ier otour i the figure be: Γ =

22 The law of the irulatory flow about a defletig body Let the irulatio aroud the ier ad outer otour be oeted alog the lie A-B. The irulatio aroud the hathed area a ow be writte a: Γ B 1 = Γ1 + Γ + d A D C

23 The law of the irulatory flow about a defletig body From the figure we a ee that: B A = D C The irulatio aroud the hathed area a ow be writte a: Γ = Γ Γ 1 1

24 The law of the irulatory flow about a defletig body Sie we do ot have ay body iide the hathed area: Γ1 = Γ1 Γ = 0 Whih give: Γ = Γ 1

25 The law of the irulatory flow about a defletig body Γ = Γ 1 Thi lead to the theorem: For a give flow oditio (with otat eergy), the irulatio aroud the defletig body i idepedet of the ize ad hape of the otour alog whih the irulatio i meaured.

26 The law of the irulatory flow about a defletig body The mea veloity for the irulatio aroud a otour havig the legth i: Γ = m = For a otat value of the irulatio, the mea veloity, m ha to dereae if the legth ireae. The irulatio i i ivere ratio to the ditae of the

27 Cirulatio about everal defletig bodie We have 3 wig profile i a twodimeioal aade ad make a otour aroud the whole aade. Thi otour i marked ABGDEF. E Γ1 = = + AEF A A E B Γ = = ABDE A D B E D A E D Γ3 = = + BGD B B D

28 Cirulatio about everal defletig bodie From the figure we a ee that: E A = A E D B = B D

29 Cirulatio about everal defletig bodie Cirulatio aroud 3 wig profile i a aade beome: A B Γ1 + Γ + Γ3 = E A D B E D Γ 1 + Γ + Γ3 = Γ0

30 Caade i a axial flow turbie Let u look at the ylidrial etio AB through the axial flow turbie.

31 Caade i a axial flow turbie By ufoldig the ylidrial etio AB from the lat lide, we a look at the blade i a aade

32 Caade i a axial flow turbie Cirulatio aroud the blade i: (where Z i the umber of blade) Γ = ZΓ i = b b + a b + a a + b a

33 Caade i a axial flow turbie From the figure we a ee that: b b = Π r u a a = Π r u1

34 Caade i a axial flow turbie Γ = ZΓ i = Π r u + a b Π r u1 + b a

35 Caade i a axial flow turbie Γ = ZΓ i = Π r u From the figure we a ee that: + a b b a Π r = a b u1 + b a

36 Caade i a axial flow turbie The irulatio beome: Γ = Π r Π r u u1

37 Caade i a axial flow turbie The hage of agular mometum i related to the vae irulatio by the equatio: Z Γ Π = r u r u1

38 Caade i a axial flow turbie By multiplyig the hage of agular mometum from the uptream to the dowtream ide of a turbie ruer i the torque atig o the turbie haft with the agular veloity of the ruer we will reogize Euler turbie equatio. E = ω r u ω r u1 = ZΓ Π E = u u u 1 u1

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