Supersonic Wings with Skin Friction: a Primer on Boundary Layers and Skin Friction

Section 8, Lecture 1 Supersonic Wings with Skin Friction: A Primer on Boundary Layers and Skin Friction

• Not in Anderson

1 MAE 5420 - Compressible Fluid Flow Lift/Drag Ratio on Symmetric Double-Wedge Airfoil

M = 3.0 • For Inviscid flow t/c = 0.0175 Supersonic Lift to drag ratio Increasing thickness ratio almost infinite for very thin airfoil

• But airfoils do not + fly in inviscid flows =

2 MAE 5420 - Compressible Fluid Flow Example: Symmetric Double-wedge

Airfoil … L/D (cont’d) • Friction effects M = 3.0 have small effect on Nozzle flow or flow in “large + “ducts” skin friction • But contribute significantly to reduce the performance of supersonic wings=

3 MAE 5420 - Compressible Fluid Flow Types of Drag

4 MAEW.H. Mason, 5420 Virginia- Compressible Tech Fluid Flow What causes “Skin Friction” • Gases are not completely non -viscous … when gas flows over a solid boundary, Molecules “stick” to the surface and impede the flow of other molecules … eventually creating an uneven velocity distribution for some small but finite layer … the “boundary layer”

5 MAE 5420 - Compressible Fluid Flow What causes “Skin Friction” (cont’d)

• The “momentum defect” within the boundary layer manifests itself as a “drag” on the surface

V d y u

6 MAE 5420 - Compressible Fluid Flow Two Types of “Boundary Layers” • Laminar • Turbulent

7 MAE 5420 - Compressible Fluid Flow Boundary Layer (continued) • Two Types of Boundary Layers

Laminar

flow

Transitional Turbulent

• Laminar --> orderly • Turbulent --> chaotic

8 MAE 5420 - Compressible Fluid Flow Laminar Boundary Layer Model

9 MAE 5420 - Compressible Fluid Flow “Skin Friction” Momentum Defect Model Start with incompressible flow for now ( r = const ) • Look at the flow near the aft end of a flat plate with span, b

• Mass flow into the small segment e dy:

• Momentum defect from freestream across segment dy:

• Assuming no pressure gradient along plate

Drag = loss in momentum

à 10 MAE 5420 - Compressible Fluid Flow Simple “Skin Friction” Model (cont’d) • Integrating across the depth of the boundary layer

“momentum thickness”11 MAE 5420 - Compressible Fluid Flow Simple “Skin Friction” Model (concluded) • Normalize Drag by plate area (bc) and dynamic pressure,

Q à “momentum thickness”

12 MAE 5420 - Compressible Fluid Flow Velocity Profile in Laminar Boundary Layer

• Simple Velocity Profile Model Laminar (parabolic)

2 u(y) ⎡ ⎛ y ⎞ ⎛ y ⎞ ⎤ = 2 ⋅a − b ⋅ ⎢ ⎝⎜ ⎠⎟ ⎝⎜ ⎠⎟ ⎥ Ve ⎣ δ δ ⎦ Pick {a, b} to Best Match Theoretical Model from Navier Stokes Equations (Blasius Model)

Fitting with “near” Velocity Profile Allows Use of Integral Methods

http://mae- Theoretical Profile: nas.eng.usu.edu/MAE_5420_Web/sec tion8/Blasius_files/Blasius.html

13 MAE 5420 - Compressible Fluid Flow Calculate Skin Friction Coefficient for Laminar Boundary Layer, Integral Method

2 u(y) ⎡ ⎛ y ⎞ ⎛ y ⎞ ⎤ y u(δ ) = 2 ⋅a − b ⋅ ... = 1→ = 1 → 2 ⋅a − b = 1→ b = 2 ⋅a −1 ⎢ ⎝⎜ ⎠⎟ ⎝⎜ ⎠⎟ ⎥ Ve ⎣ δ δ ⎦ δ Ve

2 2 δ 1 u ⎛ u ⎞ dy δ 1 ⎡ ⎛ y ⎞ ⎛ y ⎞ ⎤ ⎛ ⎡ ⎛ y ⎞ ⎛ y ⎞ ⎤⎞ dy C = 2 ⋅ ⋅ ⋅ 1− ⋅ = 2 ⋅ ⋅ 2 ⋅a − b ⋅ ⋅ 1− 2 ⋅a − b ⋅ ⋅ Dfric ⎢ ⎜ ⎟ ⎜ ⎟ ⎥ ⎢ ⎜ ⎟ ⎜ ⎟ ⎥ ∫0 ⎜ ⎟ ∫0 ⎝ ⎠ ⎝ ⎠ ⎜ ⎝ ⎠ ⎝ ⎠ ⎟ c Ve ⎝ Ve ⎠ δ c ⎣ δ δ ⎦ ⎝ ⎣ δ δ ⎦⎠ δ Let …

1 2 y δ 2 2 → ≡ ξ → CD = 2 ⋅ ⋅ ⎡2 ⋅a ⋅ξ − b ⋅ξ ⎤ − ⎡2 ⋅a ⋅ξ − b ⋅ξ ⎤ ⋅dξ δ fric c ∫0 ⎣ ⎦ ⎣ ⎦ Expand and Integrate

1 δ 2 2 3 2 4 CD = 2 ⋅ ⋅ ⎡2 ⋅a ⋅ξ − 4 ⋅a + b ⋅ξ + 4 ⋅a ⋅b ⋅ξ − b ⋅ξ ⎤⋅dξ = fric c ∫0 ⎣ ( ) ⎦ 2 3 4 5 δ ⎡ ξ 2 ξ ξ 2 ξ ⎤ 2 ⋅ ⋅ ⎢2 ⋅a ⋅ − (4 ⋅a + b)⋅ + 4 ⋅a ⋅b ⋅ − b ⋅ ⎥ c ⎣ 2 3 4 5 ⎦

⎡ 4 ⋅a2 + b b2 ⎤ δ → C = 2 ⋅ a − + a ⋅b − ⋅ Dfric ⎢ ⎥ ⎣ 3 5 ⎦ c 14 MAE 5420 - Compressible Fluid Flow Simple Skin Friction Model for Laminar Boundary Layers

⎡ 4 ⋅a2 + b b2 ⎤ δ → C = 2 ⋅ a − + a ⋅b − ⋅ Dfric ⎢ ⎥ ⎣ 3 5 ⎦ c • But what is d?

2 • Define BL thickness as the distanceu (fromy) ⎡ the⎛ platey ⎞ ⎛ y ⎞ ⎤ y u(δ ) = 2 ⋅a − b ⋅ ... = 1→ = 1 → 2 ⋅a − b = 1→ b = 2 ⋅a −1 ⎢ ⎝⎜ ⎠⎟ ⎝⎜ ⎠⎟ ⎥ where local fluid velocity reaches freeVe -stream⎣ δvelocity.δ ⎦ δ Ve • Start with Wall Shear Stress .. Shearing force per unit area ⎛ ⎞ ⎜∂u ⎟ roportional → τ = µ⋅ ⎟ wall ⎜ ⎟ ⎝∂ y⎠y=0

Constant of Proportionality

“dynamic viscosity” 15 MAE 5420 - Compressible Fluid Flow Boundary Layer Thickness ⎛∂u ⎞ → τ = µ⋅⎜ ⎟ wall ⎜ ⎟ ⎝∂ y⎠y=0

•From Earlier derivation of momentum thickness

MAE 5420 - Compressible Fluid Flow 16 Boundary Layer Thickness (cont’d)

Take derivative of above w.r.t. x à

• Classical boundary layer equation .. Von Karman Momentum law for laminar boundary layer

17 MAE 5420 - Compressible Fluid Flow Boundary Layer Thickness (cont’d) • Now Calculate Laminar Boundary Layer Shear Stress Using Velocity Distribution Model

2 ⎛ ∂u ⎞ ∂ ⎡ ⎛ y ⎞ ⎛ y ⎞ ⎤ 2 ⋅a τ wall = µ ⋅ = µ ⋅Ve ⎢2 ⋅a⎜ ⎟ − b ⋅⎜ ⎟ ⎥ = ⋅ µ ⋅Ve ⎝⎜ ∂y⎠⎟ ∂y ⎝ δ ⎠ ⎝ δ ⎠ δ y=0 ⎣ ⎦y=0

• Two ways of calculating twall

2⋅a ∂Θ τ = ⋅µ⋅V ... τ = ρ ⋅V 2 ⋅ wall δ e wall e ∂x • Equating terms … ∂Θ 2⋅a ρ ⋅V 2 ⋅ = ⋅µ⋅V e ∂x δ e ∂Θ 2⋅a ρ ⋅V ∂Θ 2⋅a ρ ⋅V 2 ⋅ = ⋅µ⋅V → e ⋅ = e ∂x δ e µ ∂x δ 18 MAE 5420 - Compressible Fluid Flow Laminar Boundary Layer Thickness (cont’d) • From earlier momentum analysis

⎛ ⎞ ⎡ 2 2 ⎤ c ∂Θ ∂ ⎜c ⎟ ⎢ 4⋅a + b b ⎥ ∂δ Θ= ⋅CD → = ⎜ ⋅CD ⎟= a− + a⋅b− ⋅ fric ⎜ fric ⎟ ⎢ ⎥ 2 ∂x ∂x⎝2 ⎠ ⎣ 3 5 ⎦ ∂x

2 ∂Θ 2⋅a ρ ⋅Ve ∂Θ 2⋅a ρ• ⋅FromVe ⋅ Previous= ⋅ µpage⋅Ve → ⋅ = ∂x δ µ ∂x δ

ρ ⋅V ⎡ 4⋅a2 + b b2 ⎤ ∂δ 2⋅a e ⋅⎢a− + a⋅b− ⎥ ⋅ = ⎢ ⎥ µ ⎣ 3 5 ⎦ ∂x δ • Separating Variables

ρ ⋅V ⎡ 4⋅a2 + b b2 ⎤ e ⋅⎢a− + a⋅b− ⎥ ⋅ ⋅∂ = 2⋅a⋅∂x ⎢ ⎥ δ δ µ ⎣ 3 5 ⎦

19 MAE 5420 - Compressible Fluid Flow Laminar Boundary Layer Thickness (cont’d)

• From earlier analysis ρ ⋅V ⎡ 4⋅a2 + b b2 ⎤ e ⋅⎢a− + a⋅b− ⎥ ⋅ ⋅∂ = 2⋅a⋅∂x ⎢ ⎥ δ δ µ ⎣ 3 5 ⎦ • Define • Integrate from {0,c}

ρ ⋅V ⎡ 4⋅a2 + b b2 ⎤ δ 2 e ⋅⎢a− + a⋅b− ⎥ ⋅ = 2⋅a⋅c ⎢ ⎥ µ ⎣ 3 5 ⎦ 2 • “Reynolds Number” • Solve for d

c 4a c 4a δ = = 1/2 2 2 1/2 2 2 ⎛ρ ⋅V ⎞ ⎡ 4⋅a + b b ⎤ R ⎡ 4⋅a + b b ⎤ ⎜ e ⎟ ⎢a− + a⋅b− ⎥ ( e ) ⎢a− + a⋅b− ⎥ ⎜ ⋅c⎟ ⎢ ⎥ ⎢ ⎥ ⎝⎜ µ ⎠⎟ ⎣ 3 5 ⎦ ⎣ 3 5 ⎦

20 MAE 5420 - Compressible Fluid Flow Laminar Boundary Layer Thickness (cont’d) • Sub into Skin Friction Model

2 2 δ 1 4a ⎡ 4 ⋅a + b b ⎤ δ = → C = 2 ⋅ a − + a ⋅b − ⋅ c 1/2 ⎡ 2 2 ⎤ Dfric R 4⋅a + b b ⎢ ⎥ ( e ) ⎢a− + a⋅b− ⎥ ⎣ 3 5 ⎦ c ⎢ ⎥ ⎣ 3 5 ⎦

⎡ 4⋅a2 + b b2 ⎤ 4a 2⋅⎢a− + a⋅b− ⎥ ⋅ ⎢ 3 5 ⎥ ⎡ 4⋅a2 + b b2 ⎤ ⎡ 4⋅a2 + b b2 ⎤ ⎣ ⎦ ⎢a− + a⋅b− ⎥ 4 a⋅⎢a− + a⋅b− ⎥ ⎢ 3 5 ⎥ ⎢ 3 5 ⎥ C = ⎣ ⎦ = ⎣ ⎦ D fric 1/2 1/2 (Re ) (Re )

δ 1 4(1) 5.477 = = 1/2 2 2 1/2 c R ⎡ 4⋅(1) +(1) (1) ⎤ R ( e ) ⎢(1)− +(1)⋅(1)− ⎥ ( e ) ⎢ ⎥ ⎣ 3 5 ⎦ If … {a,b}={1,1} ⎡ 4⋅(1)2 +(1) (1)2 ⎤ 4 (1)⋅⎢(1)− +(1)⋅(1)− ⎥ ⎢ 3 5 ⎥ 1.461 C = ⎣ ⎦ = D fric 1/2 1/2 (Re ) (Re ) 21 MAE 5420 - Compressible Fluid Flow “Best Match” Velocity Distribution Parabolic “Best Fit” Model {a, b}={1, 1} Shape Parameters “Best Fit” Parabolic Model {a, b}={1.02, 1.05} δ 1 4a 4.98 = = 1/2 ⎡ 2 2 ⎤ 1/2 c R 4⋅a + b b R ( e ) ⎢a− + a⋅b− ⎥ ( e ) Blasius Model ⎢ ⎥ c ⎣ 3 5 ⎦ http://mae- nas.eng.usu.edu/MAE_5420_Web/sec ⎡ 2 2 ⎤ ⎢ 4⋅a + b b ⎥ δ tion8/Blasius_files/Blasius.html CD = 2⋅ a− + a⋅b− ⋅ fric ⎢ ⎥ ⎣ 3 5 ⎦ c 4 ⋅4.98 δ 4 δ 15 1.328 = 0.2667⋅ ≈ = 1/2 = 1/2 c 15 c R R ( e ) ( e ) x c

22 MAE 5420 - Compressible Fluid Flow Laminar Boundary Layer (cont’d) • Approximate Drag Coefficient Model based on “Best Fit” Parabolic Velocity Profile δ 4.98 1.328 = C = 1/2 D fric 1/2 c R R ( e ) ( e ) c x

• Compares well with Exact “Blasius Solution” … slightly under predicts boundary layer height

⎛δ⎞ 5 ⎜ ⎟ = ⎜c⎟ 1/2 ⎝ ⎠Blasius R ( e ) c

• Blasius method described in Appendix to section 8 … MAE 5420 students not responsible for exact method 23 MAE 5420 - Compressible Fluid Flow Laminar Boundary Layer (Concluded) Blasius Laminar Boundary Layer Solution Summary

δ 5 = 1/2 x R ( e ) x

∂u 0.332⋅ρ ⋅Ve τwall = µ⋅ |y=0= 1/2 ∂ y R ( e ) x

C = D fric

24 MAE 5420 - Compressible Fluid Flow Comparison of Reynolds Number and Mach Number

Mach number is a measure of the ratio of the fluid • As derived in section 3 Kinetic energy to the fluid internal energy (direct motion To random thermal motion of gas molecules) -- Fundamental Parameter of Compressible Flow --

1 ρ ⋅V 2 4 e Inertial − Force / Area = ⋅ 2 ∝ • ρ c τwall Viscous− Force / Area

Reynold’s number is a measure of the ratio of Inertial Forces acting on the fluid -- to -- Viscous Forces Acting on the fluid -- Fundamental Parameter of Viscous Fluid Flow --

25 MAE 5420 - Compressible Fluid Flow What are the Units of µ, Re? • From Definition for Viscosity à

1000*Centipoise= Pa-sec • Most common table lookup form for viscosity data 26 MAE 5420 - Compressible Fluid Flow What are the Units of µ? (2) • YOU’ve Used Viscosity Units Before Motor Oil?à

A 5W-30 oil is a mult-viscosity oil that behaves as 5- 5W-30 Oil Rating? weight oil at low temperatures but gives the protection of 30-weight oil at the high engine operating temperatures. SAW J300 Oil Classification Viscosity is defined as oil’s resistance to flow and shear and “W” is for Winter! is expressed as centipoise (cP).

27 MAE 5420 - Compressible Fluid Flow Gas Viscosity Model • Sutherland’s Formula Result from kinetic theory that expresses viscosity as a function of temperature

• Write µ in terms of freestream temperature Outside of Boundary layer

28 MAE 5420 - Compressible Fluid Flow

Reynolds Number Scales?

• High-Speed and Large Configurations Operate with Mostly Turbulent Boundary Layers 9 • Aircraft Carrier – ReL ~ 10 • F-18 – 25 x 106

2 • Bumble Bee – ReL ~ 10

• Low-Speed and Small Configurations Operate with -5 Mostly Laminar Boundary • Bacterium – ReL ~ 10 Layers

29 MAE 5420 - Compressible Fluid Flow

Turbulent Boundary Layer Model

2 u(y) ⎡2 ⋅ y ⎛ y ⎞ ⎤ = − a ⋅ ⎢ ⎝⎜ ⎠⎟ ⎥ Ve ⎣ δ δ ⎦

30 MAE 5420 - Compressible Fluid Flow Velocity Profile in Turbulent Boundary Layer • Power-Law Velocity Profile Models Turbulent (exponential curve fit) We’ll find out what n is later

31 MAE 5420 - Compressible Fluid Flow Calculate Skin Friction Coefficient for Turbulent Boundary Layer • Turbulent Boundary Layer

32 MAE 5420 - Compressible Fluid Flow Turbulent Skin Friction Revisited • Turbulent Boundary Layer

2 • No Theoretical Prediction for Boundary Layer ⎛ ∂u ⎞ ∂ ⎡ ⎛ y ⎞ ⎛ y ⎞ ⎤ 2 ⋅a τ wall = µ ⋅ = µ ⋅Ve ⎢2 ⋅a⎜ ⎟ − b ⋅⎜ ⎟ ⎥ = ⋅ µ ⋅Ve ⎝⎜ ∂y⎠⎟ 0 ∂y ⎝ δ ⎠ ⎝ δ ⎠ δ Thickness for Turbulent Boundary layer y=0 ⎣ ⎦y=0

• “Time averaged” Statistical Empirical Correlation … Herrman Schlichting

33 MAE 5420 - Compressible Fluid Flow Skin Friction Revisited (Turbulent Boundary Layer) (concluded) • Turbulent Boundary Layer

34 MAE 5420 - Compressible Fluid Flow Compare Laminar and Turbulent Skin Friction Models δ 4.98 1.328 = C = 1/2 D fric 1/2 c R R ( e ) ( e ) c Laminar x

δ 0.16 = 1/n c R ( ec ) Turbulent

35 MAE 5420 - Compressible Fluid Flow Compare Laminar and Turbulent Skin Friction Coefficients with Wave Drag Coefficient δ 4.98 1.328 = C = Flat Plate with Chord “c” 1/2 D fric 1/2 c R R ( e ) ( e ) c x

Not! Independent of size

Independent of size

36 MAE 5420 - Compressible Fluid Flow Compare Laminar and Turbulent Skin

Friction (cont’d)

Plot CF Formulae

Versus Re

37 MAE 5420 - Compressible Fluid Flow Compare Laminar and Turbulent Skin

Friction (cont’d)

So Which Formula Is Correct?

38 MAE 5420 - Compressible Fluid Flow Classical Skin Friction Correlations

“Smooth Flat Plate with No Pressure Gradient” “Schoenherr Plot”

Re~10,000,000

Effect of Compressibility

Re~500,000

39 MAE 5420 - Compressible Fluid Flow

Plot the CF Laws

{a, b} = {1, 1} “Blasius exact solution for Laminar Flow”

40 MAE 5420 - Compressible Fluid Flow Boundary Layer Transition

Re~500,000 Laminar

flow

Transitional Turbulent

• Laminar --> orderly • Turbulent --> chaotic

41 MAE 5420 - Compressible Fluid Flow Bypass Vs. Natural Transition?

• Why did NASA Risk a Space Walk on STS-114 with Steve Robinson to remove Shuttle Tile Gap Filler?

• To Avoid Premature Bypass Boundary Layer Transition! During Reentry 42 MAE 5420 - Compressible Fluid Flow Bypass Vs. Natural Transition? (2)

“Transition Region”

Space Shuttle

43 MAE 5420 - Compressible Fluid Flow Bypass Vs. Natural Transition? (3)

44 MAE 5420 - Compressible Fluid Flow Bypass Vs. Natural Transition? (3) • Reynold’s Analogy … correlation of heat transfer to skin friction

• Turbulent eddy transports momentum from core flow to V fluid near wall .. It also transports heat …. u(y) dy Ue momentum heat "u ! w = µ y "y y=0

• Bypass transition dramatically Increases heat transfer to the skin surface!

45 MAE 5420 - Compressible Fluid Flow Skin Friction Drag on Transitional Boundary Layer

• Accumulated Skin Drag Proportional to Boundary Layer Thickness at Location X

46 MAE 5420 - Compressible Fluid Flow Skin Friction Drag on Transitional Boundary Layer (2) Laminar:

Turbulent:

• Empirical turbulent correlations include effects of laminar region on accumulated drag, no need for correction

What about this region?

47 MAE 5420 - Compressible Fluid Flow Skin Friction Drag on Transitional Boundary Layer (3) • “Splice” two curves together for transition region

48 MAE 5420 - Compressible Fluid Flow Skin Friction Laws Revisited

49 MAE 5420 - Compressible Fluid Flow Collected Simple Skin Friction Model (1) • So For Our Purposes … we’ll use the “1/nth power” Boundary layer law … and the Exact Laminar Solution + Transitional Splice

Re < 500,000 • See appendix on Section 8 web link for exact solution Exact Blasius Solution

500,000 < Re < 10,000,000

“Splice” of Laminar Solution and 1/7th power law Model

50 MAE 5420 - Compressible Fluid Flow Collected Simple Skin Friction Model (2)

7 8 -> n=7 10 < Re < 10

“transitions” ~ continuous

… for most turbulent flows 1/7th power law is sufficiently accurate 8 9 -> n=Log [R ] -1 10 < Re < 10 10 e

51 MAE 5420 - Compressible Fluid Flow Skin Friction Laws, The Full Monty 8 9 -> n=Log [R ] -1 10 < Re < 10 10 e

n = 1/7

52 MAE 5420 - Compressible Fluid Flow How About Three Dimensional Configurations? NACA

“Karman-Schoenherr Correlation”

53 MAE 5420 - Compressible Fluid Flow Skin Friction Laws Revisited

Karman-Schoenherr Curve Good fit To 2-D data

Effects of cross flow can cause 3-D Bodies to transition sooner than 2-D Bodies

54 MAE 5420 - Compressible Fluid Flow Questions?

55 MAE 5420 - Compressible Fluid Flow