Final Review of AeE 4 Class
Content of Aeodynamics I I Chapte : Review of Multivaiable Calculus Chapte : Review of Vectos Chapte : Review of Fluid Mechanics Chapte 4: Consevation Equations Chapte 5: Simplifications to Consevation Equations Chapte 6: Basic flows Chapte 7: Incompessible Flow Ove Aifoils
Chapte : Review of Multivaiable Calculus Review of Patial Diffeentials and Chain ules Definition of Patial Deivatives Popeties of Patial Deivatives Chain Rules
Chapte : Review of Vectos Definition of a vecto Popeties of vectos = Addition, subtaction, Scala poduct, Vecto poduct, Tiple poduct Unit vecto, vecto poduction ules, components of a vecto. Commonly used coodinate systems: Relation Between Coodinate Systems Geneal Tansfomation, invese tansfomation Scale factos, Relation between unit vectos and thei deivatives Catesian system, cylindical system, A spheical system Vecto calculus: Del, The Vecto Diffeential Opeato: Concept of Gadient Concept of Diectional deivative Concept of Divegence of a velocity fields Gauss Divegence Theoem: Gadient Theoem: eˆ h eˆ ˆ e eˆ Φ = h = h h h cs cs q + h q + h Φ eˆ ˆ Φ e q ( hh [ q + h dφ = Φ d q + A ) ( hh + q h q Φ q q ( F da) = ( F) dv CV ( ϕ d A ) = ( ϕ) dv CV = Φ ( deˆ) A ) ( hh + A ) ]
Chapte : Review of Fluid Mechanics Units and Basic definitions: System and contol volume Density, Tempetue, Pessue of fluid Steamline, path line, steak line -D, -D, -D flows Incompessible flows Steady flows Newtonian and non-newtonian fluid ideal fluid: non-heat conducting, homogenous, incompessible, inviscid fluid.
Chapte 4: Consevation Equations Methods to descibe fluid motion Lagange appoach Euleian appoach Substantial/total deivative DΦ = ( Dt DA = ( Dt + V ) Φ t + V ) A t Reynolds tanspot theoem: D D Nsys dv dv V da 4 Dt ( ) = sys CV CS 444Dt ( αρ ) = ( ) ( ) 4444 4 t αρ + α ρ 4444 44444 Consevation of mass: Integal fom: Diffeential fom: Lagangian Euleian α = ( ) + = 0 ρ dv CV ρv da t CS ρ + ρv = 0 t Dρ + ρ V = 0 Dt
Chapte 4: Consevation Equations Consevation of momentum: Integal fom: D D Nsys dv dv V da 4 Dt ( ) = sys CV CS 444Dt ( αρ ) = ( ) ( ) 4444 4 t αρ + α ρ 4444 44444 Lagangian Euleian Diffeential fom (Navie-Stokes Equations): Inviscid flow, α = V Fext enal ( ρv dv ) + V ( ρv da) = t CV CS V ρ + p τ + ρ f = 0 t V p = + f t ρ Eule equation
Chapte 5: Simplifications to Consevation Equations Benoulli's equation ideal flow (non-heat conducting homogeneous, inviscid, incompessible): A steady flow along a steamline p + ρv + ρ g z = const. A steady iotational flow By definition, the equation fo steamline will be : ds V = eˆ h dq V h eˆ V dq h eˆ V dq = 0
Chapte 5: Simplifications to Consevation Equations Voticity, ciculation, and stokes theoem ϖ = V = Voticity vecto : Ciculation: Stokes theoem: Steam function: Γ = h h h Γ = V dl Only -D, steady incompessible u = flow: h q C V C dl u = = h ˆ e q hv ψ h q S ψ h ˆ e q h V h ˆ e q h V ( V ) ds
Chapte 5: Simplifications to Consevation Equations Potential flow: Iotational+ ideal flow (non-heat conducting homogeneous, inviscid, incompessible): -D o -D -D potential flow: Laplace Equations u u u = = = φ h q φ h q φ h q Significance of intoductions of steam function and potential function: Reduce unknowns: 4 equations to equation nonlinea PDE to linea PDE Diect pocedue to solve poblem Supeposition of solutions to get a new solution Φ = ψ = 0 0
Chapte 6: Basic flows Basic flow : Unifom flow at an angle of α u = V v = V ψ = V φ = V cosα sinα sinα x V cosα x + V cos sin y y α
Chapte 6: Basic flows Basic flow : -D souce (sink) flow V = K π eˆ K ψ = θ π K φ = ln π K is the stength of the souce (o sink)
Chapte 6: Basic flows Basic flow : combination of unifom flow and a -D souce (sink) flow Stagnation point: Body shape (steamline pass the stagnation point) K is the stength of the souce (sink)
Chapte 6: Basic flows Basic flow 4:combination of a -D souce and a -D sink Rankine oval P (x,y) θ θ θ souce sink -D doublet μ sinθ ψ = π μ cosθ φ = π
Chapte 6: Basic flows Basic flow 5: unifom flow +a -D doublet = flow aound a cicula cylinde Stagnation points Pessue coefficient on the suface of cylinde R ψ = V sinθ ( ) R φ = V cosθ ( + ) R V = V cosθ ( ) R Vθ = V sinθ ( + )
Chapte 6: Basic flows Basic flow 5: -D votex flow Γ ψ = ln( ) π R Γ φ = θ π V = 0 V θ Γ = π
Chapte 6: Basic flows Basic flow 6: Flow aound a spinning cylinde = unifom flow +-D doublet + -D votex flow Stagnation points Foce on the suface of cylinde Dag Lift Kutta-Joukowski theoem L' = ρv Γ
Chapte 7: Incompessible Flow Ove Aifoils nomenclatue of aifoil angle of attack α Chod line Mean cambe line Leading edge and tailing edge Cente of pessue Aeodynamic cente
Chapte 7: Incompessible Flow Ove Aifoils Votex Sheet o votex suface theoy An aifoil can be eplaced by votex sheets located at the sufaces of the aifoil. The votex sheet is assumed to be located at the mean cambe line fo a thin aifoil
Chapte 7: Incompessible Flow Ove Aifoils Votex Sheet o votex suface γ(s) = u -u It states that the local jump in tangential velocity acoss the votex sheet is equal to the local stength of the votex sheet.
Chapte 7: Incompessible Flow Ove Aifoils Kutta condition Fo a given aifoil at a given angle of attack, the value of aound the aifoil is such that the flow leaves the tailing edge smoothly. γ V V 0 ( TE) = u L =
Chapte 7: Incompessible Flow Ove Aifoils stating votex Befoe the flow stating afte the flow stating
Chapte 7: Incompessible Flow Ove Aifoils Thin aifoil theoy: Thee assumptions Thickness of the aifoil (t/c) is small. The cambe of the aifoil (z/c) is small. Angle of Attack is small. Pinciple: A thin aifoil can be eplaced by a votex sheet located at mean cambe line. Mean cambe line is teated as a steamline. The flow velocity along the mean cambe line is tangential to the mean cambe lin. i.e. V V n = 0
Chapte 7: Incompessible Flow Ove Aifoils Thin aifoil theoy: Z V V V n n n,, Fundamental contolling equation = V n = V votex, + V = n, votex dz ( α ) dx π TE LE = γ ( s) dx x x 0 0 α V Leading Edge γ(s) ds dx x β=tan - (-dz/dx) x 0 Mean Cambe Line c α Nomal to suface point P(x 0, z 0 ) β=tan - (-dz/dx) x0 Tangential to suface X Tailing Edge π TE LE γ ( s) dz dx = V ( α ) x0 x dx
Chapte 7: Incompessible Flow Ove Aifoils Fo a symmetical thin aifoil: π Vaiable tansfomation TE γ ( x) dx x x LE 0 x = c ( cosθ ) = V α Solution: γ ( θ ) α V = ( + cosθ ) sinθ Check the Kutta condition Calculate the lift coefficient Calculate the momentum coefficient C l = πα dcl = π dα C, = C m LE Cm, c / 4 = 0 l / 4
Chapte 7: Incompessible Flow Ove Aifoils Cambeed aifoil Fo a cambeed thin aifoil Symmetical aifoil AOA C α = π ( α α L= 0 = = π L= 0 (cosθ ) dcl = π dα Cl π Cm, LE = [ + ( A A 4 4 π Cm, c = ( A A ) 4 4 πc xcp = [ c + ( A A )] 4 C A l n π π 0 π 0 l ) dz cos nθ dx dθ dz dx dθ )]