AE301 Aerodynamics I UNIT B: Theory of Aerodynamics

Transcription

1 AE301 Aeodynamics I UNIT B: Theoy of Aeodynamics ROAD MAP... B-1: Mathematics fo Aeodynamics B-2: Flow Field Repesentations B-3: Potential Flow Analysis B-4: Applications of Potential Flow Analysis AE301 Aeodynamics I : List of ubjects 3-D Coodinate ystems 2-D Catesian and Pola Coodinates Gadient of a cala Divegence and Cul Line, uface, and olume Integals Consevation Laws ubstantial Deivative Govening Equations

2 Page 1 of 11 3-D Coodinate ystems Catesian Coodinate ystem Cylindical Coodinate ystem ecto in Catesian pace: A A i A j A k x y elocity Field: ui vj wk ecto in Cylindical pace: A A e A e A e elocity Field: e e e ECTOR ADDITION AND UBTRACTION ecto addition: A B C ecto subtaction: ECTOR PRODUCT cala poduct (dot poduct): A B D => A B D A B A B cos AB A B e G ecto poduct (coss poduct): sin ˆ CARTEIAN AND CYLINDRICAL 3-D COORDINATE ecto in Catesian space: A A ˆi A ˆj A k ˆ x y ˆ i ˆ j k ˆ elocity field: ux, y, vx, y, wx, y, ecto in cylindical space: A A eˆ A eˆ A e ˆ eˆ eˆ e ˆ elocity field:,,,,,,

3 Page 2 of 11 2-D Catesian and Pola Coodinates y x ui vj e e 2-D FLOW FIELD 2-D Catesian velocity field: ux, y ˆ vx, y 2-D pola velocity field:, eˆ, e ˆ i ˆj TRANFORMATION OF COORDINATE Fom Catesian coodinates to pola coodinates: ucos vsin vcos usin 2 2 x y 1 y tan x Fom pola coodinates to Catesian coodinates: u cos sin v cos sin x cos y sin

4 Page 3 of 11 Gadient of a cala PREURE IN A FLOW FIELD Fo a 2-D Catesian coodinate system, a scala popety p (pessue) is a function of spatial coodinates. In aeodynamics, often it is impotant to undestand how the pessue changes in a cetain diection; this is commonly called, pessue gadient. Favoable pessue gadient: means that the pessue deceases in the diection of flow: dp 0 ds Advese pessue gadient: means that the pessue inceases in the diection of the flow: dp 0 ds Note: (in geneal) advese pessue gadient causes flow tansition (lamina to tubulent) as well as flow sepaation. Undestanding the pessue gadient is vey impotant in aeodynamics. PREURE GRADIENT The pessue gadient, p (gad p), is a vecto such that: Magnitude = maximum ate of change of p pe unit length of the coodinate space at the given point Diection = diection of the maximum ate of change of p at the given point Using pessue gadient, diectional change of pessue (this is commonly undestood as pessue gadient in the diection of the flow ) can be given: dp p ˆ ds n (This is how much change of pessue takes place in the diection specified by the unit vecto ˆn ) PREURE GRADIENT (MATHEMATICAL EXPREION) p In 3-D Catesian / cylindical coodinates: ˆ p ˆ p p i j k ˆ / x y p 1 p p p eˆ ˆ ˆ e e

5 Page 4 of 11 THE DEL OPERATOR Mathematically, the del is an opeato in vecto calculous (also called the nabla ). Taking the del, (often called gadient o gad ) a scala popety will tun into a vecto popety: x y x y p p p p gad p i j k (this is pessue gadient in Catesian) x y 1 1,, eˆ ˆ ˆ e e 1 gad p ˆ p p p e ˆ p ˆ e e (this is pessue gadient in cylindical) Catesian:,, ˆi ˆj k ˆ Fo example: ˆ ˆ ˆ Cylindical: Fo example: DIERGENCE OF A ELOCITY FIELD The dot poduct between the del opeato and the velocity (vecto popety) of a flow field is called the divegence, often denoted by div : u v w x y 1 1 div Catesian: div Divegence and Cul ui vj wk Cylindical: CURL OF A ELOCITY FIELD The coss poduct between the del opeato and the velocity of a flow field is called the cul : ˆi ˆj kˆ w v ˆ u w ˆ v u ˆ i j k x y y x x y Catesian: cul Cylindical: cul u v w eˆ eˆ eˆ ˆ ˆ ˆ e e e

6 Page 5 of 11 Class Example Poblem B-1-1 Related ubjects... Review of ecto Algeba Detemine the following vecto elations: ( ) ( A) Also, define the following impotant elementay vecto elations: () ( A) ( A) These ae so called non-tivial vecto algeba calculation ules : cul gad "eo" (vecto) and div cul A "eo" (scala) ˆ ˆ ˆ i j k x y ˆi ˆj kˆ ˆ ˆ ˆ i j k 0 x y y y x x xy yx x y ˆ A A y ˆAx A ˆ Ay A x A i j k y x x y A A A y Ax A y Ax x y y x x y A A y A A x A y Ax 0 xy x y xy x y These ae so called distibution ules in vecto algeba: A A A A A A

7 Page 6 of 11 Line, uface, and olume Integals C Ads A d LINE, URFACE, AND OLUME INTEGRAL Line integals (closed loop): A d s C ds : Line ecto = Diection tangent to the line / Magnitude = ds uface integals (closed suface): pd, A d, A d d : Aea ecto = Diection nomal to the suface / Magnitude = d olume integals: d, A d RELATION BETWEEN LINE, URFACE, AND OLUME INTEGRAL tokes theoem (tansfomation fom line => aea integal fo a vecto popety): Ads A d C Divegence theoem (tansfomation fom aea => volume integal fo a vecto popety): Ad A d Gadient theoem (tansfomation fom aea => volume integal fo a scala popety): p d p d

8 Page 7 of 11 Consevation Laws (1) CONTINUITY EQUATION IN INTEGRAL FORM tating fom the integal fom of continuity equation: dv d 0 t v Tansfom the govening equation: Fom integal fom to diffeential fom of continuity equation CONTINUITY EQUATION IN DIFFERENTIAL FORM Applying the divegence theoem: d d v ubstituting this into the integal fom of continuity equation yields: d d 0 t => d d 0 t v v v v Theefoe, d 0 t v o, simply the integand must be equal to eo, theefoe: 0 (diffeential fom of continuity equation) t

9 Page 8 of 11 Consevation Laws (2) u t v t MOMENTUM EQUATION IN INTEGRAL FORM tating fom the x-component of momentum equation: ud d u p d fxd F t x v v w t p x u fx fx p y v f y f y p w f f Tansfom the govening equation: Fom integal fom to diffeential fom of momentum equation MOMENTUM EQUATION IN DIFFERENTIAL FORM Applying the divegence and gadient theoem: p ud u d d fxd fx d t x v v v v v Theefoe, u p u fx fx d 0 t x v o, simply the integand must be equal to eo, theefoe: u p u fx fx 0 t x u p => u fx fx t x (diffeential fom of x-momentum equation)

10 Page 9 of 11 ubstantial Deivative UBTANTIAL DERIATIE Flow field in aeodynamics usually means that thee is convection. This means that the flow field popeties (both scala and vecto) ae tanspoted due to the pesence of convection (the velocity distibution within the flow field). In ode to undestand the whole pictue of flow field, theefoe, one needs to keep tack of two distinctively diffeent ates of changes within the flow field... (i) how the popety changes with espect of time at each location ( time ate of change at each local location) and (ii) how the popety changes with espect to location at each instant of time ( position ate of change due to the pesence of convection ). Note that, fo steady flow field, the time ate of change becomes eo. The TOTAL ate of change (both time and position ) of a fluid popety (fo example, an acceleation field) can be expessed mathematically in substantial (o often called, TOTAL ) deivative. In 3-D Catesian coodinate system: D Dt t t x y u v w t : time ate of change at a fixed point = local deivative u v w x y : position ate of change due to convection = convective deivative Fo example, acceleation field: D u v w a Dt t t x y local acceleation convective acceleation

11 Page 10 of 11 CONTINUITY EQUATION IN DIFFERENTIAL FORM tating fom the diffeential fom of the continuity equation: 0 t Tansfom the govening equation: Fom diffeential fom to substantial deivative fom of continuity equation CONTINUITY EQUATION IN UBTANTIAL DERIATIE FORM Using the vecto identity, Thus, 0 t Note that the fist 2 tems defines the substantial deivative ( D Dt ), that is: 0 t Theefoe, in tems of substantial deivative, continuity equation can be witten as: D 0 (continuity equation: substantial deivative fom) Dt MOMENTUM EQUATION IN UBTANTIAL DERIATIE FORM Fom diffeential fom to substantial deivative fom of momentum equations t v u t w t Govening Equations p x p v f y f y u fx fx Du p f Dt x Dv p fy Dt y => x x y => fy p w f f Du p fx Dt x Dv p fy Dt y Dw p f Dt Dw p f Dt => f f f f x y f

12 Page 11 of 11 Class Example Poblem B-1-2 Related ubjects... Govening Equations tating fom the diffeential fom of the momentum equation (x-component): u t p x u fx fx deive the diffeential fom of the momentum equation in tems of substantial deivative (x-component): Du Dt p fx x f x u p tating fom the x-momentum equation: u fx fx t x u u The fist tem (LH) can be expanded as: u t t t The second tem (LH) can also be expanded by vecto identity: u u u u ubstituting these into the momentum equation yields: u p u u u fx f x t t x u p o, u u fx fx t t x Fom continuity equation, 0, theefoe: t u p u fx fx t x u p => u f x f x t x p u f x f x t x Du p In tems of substantial deivative as: fx fx Dt x