Aerodynamics I Compressible flow past an airfoil

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Amelia Krzemińska
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1 Aerodynamics I Compressible flow past an airfoil transonic flow past the RAE-8 airfoil (M = 0.73, Re = , α = 3.19 )

2 Potential equation in compressible flows

3 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

4 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)

5 Full potential theory After transformation equation (1.10) takes following form: [ 1 1 c ( Φ x ) ] [ ( ) ] Φ x 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.

6 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)

7 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)

8 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.

9 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)

10 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)

11 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.

12 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)

13 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)

14 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)

15 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)

16 Other high order rules 1.5 Prandtl-Glauert Karman-Tsien Laitone Cp M

17 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.

18 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)

19 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

20 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)

21 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.

22 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 = α M 1 = Exact theory based on the oblique shock wave and Prandtl-Meyer expansion: C L = C D =

23 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 Prandtl-Glauert Ackeret dcl/dα M

24 Transonic flow past a D airfoil

25 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)

26 Critical Mach number For izentropic flow: Using (1.4): ( p = p p 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.

27 Critical pressure coefficient

28 Critical pressure coefficient

29 Krytyczny współczynnik ciśnienia 1.5 C p C p (PG theory) Cp M M cr

30 Critical pressure coefficient 1.5 C p C p (PG theory) Cp M

31 Critical pressure coefficient airfoil thickness

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