Aerodynamics I Compressible flow past an airfoil transonic flow past the RAE-8 airfoil (M = 0.73, Re = 6.5 10 6, α = 3.19 )
Potential equation in compressible flows
Full potential theory Let us introduce a velocity potential (no vorticity is present v = 0): v = Φ D Continuity equation can be transformed: u = Φ x v = Φ y (1.1) (ρ v) = ρ v + v ρ = 0 ρ Φ + Φ ρ = 0 (1.) From momentum equation: ( ) 1 V ρ p = v v = + ( v) v (1.3) Since vorticity v is zero : ( v) v = 0 dp = ρ d ( V ) = ρ d ( Φ Φ) (1.4) Using equation for speed of sound (izentropic relation): dp = c dρ dρ = ρ d ( Φ Φ) (1.5) c
Full potential theory Gradient of density can be obtained fro equation: ρ = ρ c ( Φ Φ) = ρ Φ Φ (1.6) c for D: [ ] ρ x = ρ Φ Φ c x x + Φ Φ y x y ρ y = ρ c after substituting into (1.): for D: Φ x + Φ y 1 c [ Φ x Φ x y + Φ y ] Φ y (1.7) (1.8) ρ Φ ρ Φ ( Φ Φ) = 0 (1.9) c [ ( ) ( ) Φ Φ Φ x x + Φ y y + Φ Φ x y ] Φ = 0 x y (1.10)
Full potential theory After transformation equation (1.10) takes following form: [ 1 1 c ( Φ x ) ] [ ( ) ] Φ x + 1 1 Φ Φ c y y = Φ c x Φ y Φ x y (1.11) Using energy integral the relation for speed of sound can be obtained: c = c 0 k 1 [ ( ) Φ + x ( ) ] Φ y (1.1) Equations (1.11) and (1.1) are equation of full potential theory. They allow to simulate compressible flows with weak (izentropic relations are used!) shock waves.
Linearized equations of small disturbances potential Equations (1.11) and (1.1) can be further simplified by introducing potential of small disturbances and using linearization. where: v = v + ṽ (1.13) v = [ u, v ], v = [ V, 0 ], ṽ = [ ũ, ṽ ] Let us define potential of disturbances φ : ṽ = φ Φ = V x + φ (1.14) After substituting into equation (1.11): [ ( ) ] [ c V + φ ( ) ] φ x x + c φ ( φ y y = V + φ x ) φ y φ x y (1.15)
Linearized equations of small disturbances potential In order to simplify the (1.15) it is convenient to use form based on velocity not the potential: [ c ( V + ũ ) ] [ ] ũ x + c ṽ ṽ ũ = (V + ũ) ṽ (1.16) y y Equation (1.1) takes form: c = c k 1 ( V ũ + ũ + ṽ ) (1.17) After substitution (1.17) into (1.16) and transformation: [ ( ) 1 M ũ x + ṽ y = M (k + 1) ũ + k + 1 ũ V V [ + M (k + 1) ṽ + k + 1 ṽ V V [ ( + M ṽ 1 + ũ ) ( ũ V V x + ṽ x + k 1 + k 1 )] ṽ V ũ V ] ũ x ] ṽ y (1.18)
Linearized equations of small disturbances potential Assuming small disturbances (valid for thin airfoils at small angles of attack): ũ V 1 ṽ V 1 (1.19) Equation (1.18) can be transformed to a form where only linear terms are present (it is assumed that it is valid for M < 0.8 and 1. < M < 5): ( ) 1 M ũ x + ṽ y = 0 (1.0) After substitution of the potential function: ( 1 M ) φ x + φ y = 0 (1.1) This equation is elliptic for M < 1 and hyperbolic for M > 1.
Pressure coefficient for linearized equations Pressure coefficient is defined by: C p p p gdzie: q = ρ V q (1.) Equation (1.) can be transformed using: q = ρ V = k p ( ρ ) V = k k p p M (1.3) The formula for pressure coefficients takes then form: ( ) C p = p 1 k M p For simplifications we will use izentropic relation:: ( ) k p T k 1 = p T (1.4) (1.5)
From eneregy integral: Pressure coefficient for linearized equations V + cpt = V k 1 ( + cpt T T = V V ) (1.6) k R Using definition of the disturbance velocity (1.13) we can obtain: ( ) T = 1 k 1 M ũ + ũ + ṽ T V V (1.7) After substituting: p p = [ 1 k 1 M ( )] ũ + ũ + ṽ k k 1 V V (1.8) It can be simplified by expanding using Taylor series and discarding higher order terms: p = (1 r) k 1 k = 1 k r +... p k 1 (1.9) ( ) p 1 k p M ũ + ũ + ṽ V V (1.30)
Pressure coefficient for linearized equations Substituting (1.30) inot (1.4): C p = ũ V ũ + ṽ V (1.31) For small disturbances (1.19): ũ + ṽ V ũ V (1.3) Finally, we can obtain the simplified relation for pressure coefficient: C p = ũ V (1.33) The relations are valid for both, subsonic and supersonic flows.
Prandtl Glauert rule Let us consider subsonic flow (M < 1) which is described by linearized equations of small disturbances potential. We can defined the coefficient: β 1 M (1.34) equation (1.1) can be then written in form: β φ x + φ y = 0 (1.35) This equation can be transformed to Laplace equation by transformation applied to the reference coordinates, e.g.: ξ = x η = βy x = ξ y = β (1.36) η After substituting into (1.35): β ϕ ξ + β ϕ η = 0 ϕ ξ + ϕ η = 0 (1.37)
Prandtl Glauert rule What is a difference between φ and ϕ? Let us assume that the boundary of the airfoil on the plane x y is defined by function y = f(x). Boundary condition for the airfoil (normal velocity is equal zero) then can be defined: d dx f(x) = In similar way for ξ η: d dξ ˆf(ξ) = ṽ V + ũ ˆv V + û ṽ ṽ = φ V y = V ˆv ˆv = ϕ V η = V Assuming that airfoils in x y and ξ η are similar then ˆf = f. d f(x) (1.38) dx d dξ ˆf(ξ) (1.39) ϕ η = 1 ϕ β y = φ y ϕ = β φ (1.40)
Prandtl Glauert rule y compressible flow φ C p, C L, C M incompressible flow η = βy ϕ = β φ Ĉ p, Ĉ L, Ĉ M y = f(x) x η = f(ξ) ξ = x Using (1.33) it is possible to obtain relation for C p: C p = ũ = 1 φ V V x = 1 ϕ β V x = û = 1 β V β Ĉp (1.41) Since moment and lift coefficients are integrals of C p finally we can obtain: C p = Ĉ p 1 M C L = Ĉ L 1 M C M = Ĉ M 1 M (1.4)
Other high order rules Karman Tsien rule: C p = 1 M + Ĉ p 1+ M 1 M Ĉ p (1.43) Laitone rule: Ĉ p C p = 1 M + M k 1 (1+ M ) 1 M Ĉ p (1.44)
Other high order rules 1.5 Prandtl-Glauert Karman-Tsien Laitone Cp 1 0.5 0 0 0. 0.4 0.6 0.8 1 M
Linearized supersonic flow In supersonic flow (M > 1) the equation (1.1) can be written: λ φ x φ y = 0 where: λ = M 1 (1.45) This is the wave equation where the wave propagation speed is equal λ. The general solution to this problem has form: φ = f +(x + λ y) + f (x λ y) (1.46) where f + i f are arbitrarily chosen functions. The values of f + and f are constant along lines defined by: x + λ y = const x λ y = const (1.47) Such lines are called characteristics of the wave equation. the angle of inclination of the characteristics can be obtained from (1.46): dy dx = ± 1 λ = ± 1 M 1 = tg(µ± ) (1.48) Characteristics of the equation (1.45) are identical to the Mach lines.
Linearized supersonic flow M > 1 µ θ The boundary condition for zero normal velocity must be satisfied on the boundary of the domain or airfoil : ṽ tg(θ) = V + ũ ṽ (1.49) V Using general solution to the wave equation (1.46): ũ = φ x = f + + f ṽ = φ y = λ (f + f ) [ ] ũ = ṽ f + + f λ f + f (1.50)
Linearized supersonic flow Using boundary condition (1.49) we can obtain: ṽ = V tg(θ) ũ V θ λ After substituting (1.51) into (1.33): C p = θ M 1 [ ] f + + f f + f [ ] f + + f f + f (1.51) (1.5) Functions f + and f can be chosen arbitrarily. In order to simplify the problem we will consider only two cases: f + is constant or f is constant. Since characteristics are identical to Mach lines their type (sign) should be chosen such that the disturbances will be propagated with the flow. Cp U/L θ = ± M 1 (1.53) wher: U upper airfoil surface, L lower airfoil surface
Linearized supersonic flow - flat plate C p upper surface α µ x µ + lower surface Upper surface characteristic µ and θ = α : C U p = α M 1 (1.54) Lower surface characteristic µ + and θ = α : C L p = α M 1 (1.55)
Linearized supersonic flow - flat plate Force coefficients in normal and tangent directions: Lift coefficient: C n = 1 lac (Cp L C U 4 α p ) dx = l ac M 1 C t = 0 0 C L = C n cos(α) C t sin(α) C n C L = 4 α M 1 (1.56) (1.57) Drag coefficient: C D = C n sin(α) + C t cos(α) C n α C D = 4 α M 1 (1.58) In potential supersonic flow, there exists a nonzero drag force (in contrary to subsonic flow s Alembert paradox). This force is called wave drag.
Linearized supersonic flow - flat plate Example: Flow past a flat plate for α = 10, M = and p = 1 Simplified theory - linearized supersonic flow: C L = C D = 4 α M 1 = 0.403 4 α M 1 = 0.0703 Exact theory based on the oblique shock wave and Prandtl-Meyer expansion: C L = 0.408 C D = 0.0719
Linearized supersonic flow - dc L /dα Incompressible flows: dc L/dα = π Subsonic flows (P-G rule): dc L/dα = π/ M 1 Supersonic flows: dc L/dα = 4 / M 1 0 15 Prandtl-Glauert Ackeret dcl/dα 10 5 0 0 0.5 1 1.5.5 3 M
Transonic flow past a D airfoil
Critical Mach number Critical Mach number M cr is a Mach number of undisturbed flow M for which maximum of the local velocity at some point on the airfoil surface is equal to the speed of sound. If M < M cr then flow is fully subsonic (a i b) If M > M cr then there exists a supersonic flow region (d)
Critical Mach number For izentropic flow: Using (1.4): ( p = p p 0 1 + k 1 = M ) k k 1 p p 0 p 1 + k 1 M [ ( C p = 1 + k 1 M ) k ] k 1 k M 1 + k 1 M 1 where M is local Mach number at some point on airfoil surface. If M = 1 then: [ ( Cp = 1 + k 1 M ) k ] k 1 1 k M 1 + k 1 (.1) (.) (.3) Cp is a critical pressure coefficient. If at some point on airfoil surface C p > Cp then at this point M > 1, otherwise if C p < Cp then M < 1.
Critical pressure coefficient
Critical pressure coefficient
Krytyczny współczynnik ciśnienia 1.5 C p C p (PG theory) Cp 1 0.5 0 0 0. 0.4 0.6 0.8 1 M M cr
Critical pressure coefficient 1.5 C p C p (PG theory) Cp 1 0.5 0 0 0. 0.4 0.6 0.8 1 M
Critical pressure coefficient airfoil thickness