HCMC University of Technology 13/10/200957:020 Fluid Mechanics
External Viscous Flow
Space Shuttle Re-entry
Planetary Boundary Layer
Automobile
HCMC University of Technology 13/10/200957:020 Fluid Mechanics
Part A – Boundary Layer
Part B – Drag & Lift
External Viscous Flow
Boundary-Layer Concept
Boundary – Layer Thicknesses
Laminar Flat-Plate Boundary Layer
Turbulent Flow
Flow over a Flat Plate Parallel to the Flow: Friction Drag
Flow over a Flat Plate Normal to the Flow: Pressure Drag
Flow over a Sphere and Cylinder: Friction & Pressure Drag
Streamlining
HCMC University of Technology 13/10/200957:020 Fluid Mechanics
Quantify the behavior of viscous, incompressible fluids in external flow
A number of phenomena occurs in the external flow over an airfoil
External Viscous Flow
HCMC University of Technology 13/10/200957:020 Fluid Mechanics
Part A: BOUNDARY LAYERS
A.1.Boundary Layer Concept
(Ludwig Prandtl – German aerodynamicist, 1904)
Viscous flows can be analyzed by dividing the flow into two regions, one close to solid boundaries, the other covering the rest of flow.
Boundary-layer concept permits the solution of viscous flow problems that would have been impossible through application of the Navier-Stokes equations
Rex > 5x105
HCMC University of Technology 13/10/200957:020 Fluid Mechanics
A.2. Boundary-layer thicknesses
Part A: BOUNDARY LAYERS
Disturbance thickness (δ): the distance from the surface at which the velocity is within 1% of the free stream, u ~ 0.99U
Displacement thickness (δ*): the distance the plate would be moved upward so that the loss of mass flux is equivalent to the loss the boundary layer causes
HCMC University of Technology 13/10/200957:020 Fluid Mechanics
A.2. Boundary-layer thicknesses
Part A: BOUNDARY LAYERS
Momentum thickness (θ): the distance the plate would be moved upward so that the loss of momentum flux is equivalent to the loss the boundary layer causes
δ*, θ are define in terms of integrals for which the integrand vanishes in the free-stream
HCMC University of Technology 13/10/200957:020 Fluid Mechanics
Part A: BOUNDARY LAYERS
Basic Assumptions for Analyses of the Boundary-layer Development:
1. u→ U at y = δ
2. δu/δy → 0 at y = δ
3. u << U within the boundary layer
4. Pressure variation across the thin boundary layer is negligible. The free-stream pressure distribution is impressed on the boundary layer
HCMC University of Technology 13/10/200957:020 Fluid Mechanics
Part A: BOUNDARY LAYERS
Example 9.1: Flow of standard air in laboratory wind tunnel. Test section in L = 305mm square. Displacement thicknesses are δ*1 = 1.5mm and δ*2 = 2.1mm. Free-stream speed is U1 = 26m/s.
Find: Change in static pressure between section 1 and 2.
HCMC University of Technology 13/10/200957:020 Fluid Mechanics
Part A: BOUNDARY LAYERS
Example 9.1:
Bernoulli Eq.
Continuity Eq.
HCMC University of Technology 13/10/200957:020 Fluid Mechanics
A.3. LAMINAR FLAT-PLATE BOUNDARY LAYER
Momentum Integral Equation
To obtain approximate information on boundary-layer growth for the general case (laminar or turbulent BL).
Determine the BL thickness, δ, as a function of x? wall shear stress, τw?
Using continuity equation and momentum equation for a control volume abcd
HCMC University of Technology 13/10/200957:020 Fluid Mechanics
A.3. LAMINAR FLAT-PLATE BOUNDARY LAYER
Continuity Equation
HCMC University of Technology 13/10/200957:020 Fluid Mechanics
A.3. LAMINAR FLAT-PLATE BOUNDARY LAYER
Momentum Equation
HCMC University of Technology 13/10/200957:020 Fluid Mechanics
A.3. LAMINAR FLAT-PLATE BOUNDARY LAYER
Forces = Pressure Forces + Viscous Shear Forces
HCMC University of Technology 13/10/200957:020 Fluid Mechanics
Momentum Integral Equation
Bernoulli Equation
Momentum Integral Equation for Either a Laminar or Turbulent BL Flow:
A.3. LAMINAR FLAT-PLATE BOUNDARY LAYER
HCMC University of Technology 13/10/200957:020 Fluid Mechanics
A.3. LAMINAR FLAT-PLATE BOUNDARY LAYER
Flat plate (zero pressure gradient): in free-stream, p = const, U = const
Momentum Integral Equation
Velocity Distribution Assumption:
with BCs:
HCMC University of Technology 13/10/200957:020 Fluid Mechanics
A.3. LAMINAR FLAT-PLATE BOUNDARY LAYER
Polynomial Velocity Profile Laminar Flow:
Boundary conditions:
a, b, c
Skin friction coefficient:
HCMC University of Technology 13/10/200957:020 Fluid Mechanics
LAMINAR FLAT-PLATE BOUNDARY LAYER
Polynomial Velocity Profile Laminar Flow:
At transition, Rex = 5x105, U = 30m/s. BL thickness at x = 0.24m:
HCMC University of Technology 13/10/200957:020 Fluid Mechanics
LAMINAR FLAT-PLATE BOUNDARY LAYER
Example 9.3: Sinusoidal Velocity Profile Laminar Flow
2D, laminar BL flow along a flat plate. BL velocity profile is
Find:
(a) δ(x), δ*(x) (b) Total friction force on a plate
HCMC University of Technology 13/10/200957:020 Fluid Mechanics
LAMINAR FLAT-PLATE BOUNDARY LAYER
Example 9.3: Sinusoidal Velocity Profile Laminar Flow
(Newtonian Shear Stress)
(Momentum Integral Equation)
(Disturbance Thickness)
HCMC University of Technology 13/10/200957:020 Fluid Mechanics
LAMINAR FLAT-PLATE BOUNDARY LAYER
Example 9.3: Sinusoidal Velocity Profile Laminar Flow
Displacement Thickness
Total Friction Force
HCMC University of Technology 13/10/200957:020 Fluid Mechanics
A.4. TURBULENT FLOW
An acceptable velocity profile for turbulent BL on smooth plates is the empirical power-law profile.
For turbulent BL flow we adapt the expression developed for pipe flow
Momentum Integral Equation:
Skin friction coefficient
Well for 5x105 HCMC University of Technology 13/10/200957:020 Fluid Mechanics
A.4. TURBULENT FLOW
Example 9.4:
Flat-plate BL flow; turbulent flow from the LE; 1/7 power velocity profile.
Find: δL; δ*L ; τw(L). Comparison with results for laminar flow from LE
HCMC University of Technology 13/10/200957:020 Fluid Mechanics
A.4. TURBULENT FLOW
HCMC University of Technology 13/10/200957:020 Fluid Mechanics
A.4. TURBULENT FLOW
Example 9.4:
Find: δL; δ*L ; τw(L). Comparison with results for laminar flow from LE
HCMC University of Technology 13/10/200957:020 Fluid Mechanics
A.5. PRESSURE GRADIENTS IN B.L FLOW
Flow separation:
Adverse pressure gradient in which pressure increases in the flow direction (δp/ δx>0) will tend to contribute to the slowing of the fluid particles. If the adverse pressure gradient is server enough, flow separation & wake occur
HCMC University of Technology 13/10/200957:020 Fluid Mechanics
A.5. PRESSURE GRADIENTS IN B.L FLOW
Flow separation
Turbulent layer is better able to resist separation in an adverse pressure gradient than laminar layer
HCMC University of Technology 13/10/200957:020 Fluid Mechanics
A.5. PRESSURE GRADIENTS IN B.L FLOW
Adverse pressure gradients cause significant changes in velocity profiles for both laminar and turbulent B.L flows.
H = δ*/ θ = velocity-profile “shape factor”
Turbulent B.L flow: H increases from 1.3 (zero pressure gradient) to 2.5 at separation
Laminar B.L flow: H increases from 2.6 (zero pressure gradient) to 3.5 at separation
Velocity distributions dU/dx over bodies must be known before eq.9.28 can be applied. By using: potential flow theory/numerical source (vortex) panel methods…
HCMC University of Technology 13/10/200957:020 Fluid Mechanics
PART B. FLUID FLOW OVER IMMERSE BODIES
Surface stresses
Tangential stresses
Normal stresses
Net forces
Given the shape of the body, compute the pressure distribution. Then integrate the pressure over the body surface → pressure forces
Use this pressure distribution to find the surface viscous stress τw. Integrate the viscous stress over the body surface → friction forces
(friction forces)
(pressure forces)
This procedure in practice is quite difficult (body shapes, flow separation, wake…)
→ experimental methods to determine the net force for most body shapes
Net force F
Drag force FD
Lift force FL
HCMC University of Technology 13/10/200957:020 Fluid Mechanics
B.1. DRAG
Dimensional Analysis (Buckingham theorem)
Drag Coefficient, CD
For compressibility or free-surface effects:
HCMC University of Technology 13/10/200957:020 Fluid Mechanics
B.1. DRAG
Flow over a Flat Plate Parallel to the Flow: Friction Drag
Total drag is equal to the friction drag:
Laminar B.L flow:
HCMC University of Technology 13/10/200957:020 Fluid Mechanics
B.1. DRAG
Flow over a Flat Plate Parallel to the Flow: Friction Drag
Turbulence B.L flow:
For B.L that is initially laminar and undergoes transition at some location on the plate, the turbulent drag coefficient must be adjusted to account for the laminar flow over the initial length
(B = 1740)
HCMC University of Technology 13/10/200957:020 Fluid Mechanics
B.1. DRAG
Flow over a Flat Plate Parallel to the Flow: Friction Drag
For ReL < 109, Schlichting eq.
HCMC University of Technology 13/10/200957:020 Fluid Mechanics
B.1. DRAG
Example 9.5: Skin Friction Drag on a Supertanker
Given: Supertanker cruising at U = 13kt
Find: (a) Force
(b) Power required to overcome skin friction drag
Solution:
Model the tanker hull as a flat plate, of length L and width b = B+2D
HCMC University of Technology 13/10/200957:020 Fluid Mechanics
B.1. DRAG (Example 9.5)
HCMC University of Technology 13/10/200957:020 Fluid Mechanics
B.1. DRAG
Flow over a Flat Plate Normal to the Flow: Friction Drag
Total drag is equal to the pressure drag:
Pressure over the rear surface of the plate is essentially constant, its magnitude cannot be determined analytically -> experiments
HCMC University of Technology 13/10/200957:020 Fluid Mechanics
B.1. DRAG
Flow over a Flat Plate Normal to the Flow: Friction Drag
Drag coefficient for flow over an immersed object usually is based on the frontal area (projected area) of the object
Drag coefficient depends on aspect ratio b/h and on the Reynold number. For Re greater than about 1000, CD is essentially independent of Re
HCMC University of Technology 13/10/200957:020 Fluid Mechanics
Flow over a Flat Plate Normal to the Flow: Friction Drag
HCMC University of Technology 13/10/200957:020 Fluid Mechanics
B.1. DRAG
Flow over a Sphere : Friction and Pressure Drag
HCMC University of Technology 13/10/200957:020 Fluid Mechanics
B.1. DRAG
Flow over a Sphere : Friction and Pressure Drag
Re<1.0: Stokes Analytically Drag Coefficient CD = 24/Re corresponds to FD ~ V
1.0103Re>3x105: transition occurs, B.L on the forward portion of the sphere becomes turbulent. The point of separation moves downstream from the sphere midsection, the size of the wake decreases. Net pressure force on the sphere is reduced, CD decreases abruptly.

HCMC University of Technology 13/10/200957:020 Fluid Mechanics
B.1. DRAG
Flow over a Sphere : Friction and Pressure Drag
A turbulent B.L has more momentum flux than a laminar B.L
→ better resist an adverse pressure gradient
→ delay separation and
→ reduces the pressure drag
Application: “dimples” on a golf ball
HCMC University of Technology 13/10/200957:020 Fluid Mechanics
B.1. DRAG
Flow over a Cylinder: Friction and Pressure Drag
HCMC University of Technology 13/10/200957:020 Fluid Mechanics
B.1. DRAG
Flow over a Cylinder: Friction and Pressure Drag
Flow about a smooth circular cylinder may develop alternating vortices downstream. The vortex shedding causes an oscillatory lift force on the cylinder perpendicular to the stream motion. Re = 60-5000. Appl: wire singing, ‘slap’ ropes
Strouhal number: St = f D/V
St = 0.21
HCMC University of Technology 13/10/200957:020 Fluid Mechanics
B.1. DRAG
Example 9.6: Aerodynamic Drag and Moment on a Chimney
Given: Cylindrical chimney, D = 1m, L = 25m, in uniform flow with V = 50km/h, p = 101kPa, T=15C
Neglect end effects
Find: Bending moment moment at bottom of chimney
HCMC University of Technology 13/10/200957:020 Fluid Mechanics
B.1. DRAG
Example 9.6: Aerodynamic Drag and Moment on a Chimney
Example 9.7: Deceleration of an Automobile by a Drag Parachute
GIVEN: Dragster weighing 1600lbf, moving with initial speed Vo = 270mph, is slowed by the drag force on a chute of area A = 25ft2.
FIND: Time required for the machine to decelerate to 100mph
HCMC University of Technology 13/10/200957:020 Fluid Mechanics
B.1. DRAG
Example 9.7: Deceleration of an Automobile by a Drag Parachute
HCMC University of Technology 13/10/200957:020 Fluid Mechanics
B.1. DRAG
Example 9.7: Deceleration of an Automobile by a Drag Parachute
HCMC University of Technology 13/10/200957:020 Fluid Mechanics
B.2. STREAMLINING
Streamlining: make the rear of the body more tapered → reduce the adverse pressure gradient and hence make the turbulent wake smaller
HCMC University of Technology 13/10/200957:020 Fluid Mechanics
B.2. STREAMLINING
HCMC University of Technology 13/10/200957:020 Fluid Mechanics
B.2. STREAMLINING
Recent advances have made possible development of low-drag shapes even better than NACA 60-series shapes
With favorable drag characteristics, laminar-flow airfoils are used in the design of most modern subsonic aircraft
Reduction of aerodynamics drag also is important for road vehicle applications. Studies on buses have shown that drag reductions up to 25percent are possible with careful attention to front contour.
HCMC University of Technology 13/10/200957:020 Fluid Mechanics
B.3. LIFT
Lift Coefficient:
HCMC University of Technology 13/10/200957:020 Fluid Mechanics
B.3. LIFT
Lift-drag polars – plots of CL versus CD – often are used to present airfoil data. The lift/drag ratio, CL/CD is very important in the aircraft design.
HCMC University of Technology 13/10/200957:020 Fluid Mechanics
B.3. LIFT
Trailing vortex causes pressure distribution reduce, leading less lift
(finite wing)
Trailing vortex can be very strong and persistent, possibly being a hazard to other aircraft for 5 to 10 miles behind a large airplane
Induced velocity makes the effective angle of attack reduce → reduce lift L. It also causes induced drag → increase drag D
HCMC University of Technology 13/10/200957:020 Fluid Mechanics
Trailing vortex
B.3. LIFT-Applications
HCMC University of Technology 13/10/200957:020 Fluid Mechanics
B.3. LIFT-Applications
Aspect ratio:
Albatross
Sailplane
Loss of lift and increase in drag caused by finite-span effects correlated with aspect ratio
(High-performance sailplane: ar = 40 with L/D = 40, typical light airplane: ar = 12 with L/D = 20, Boeing 777: ar = 8.68 with L/D = 19.5)
HCMC University of Technology 13/10/200957:020 Fluid Mechanics
B.3. LIFT-Applications
American Bald Eagle
F15
Birds that must maneuver quickly to catch their prey, such as eagles, have wings of relatively short span, but large area, which gives low wing loading (ratio of weight to platform area) and thus high maneuverability
HCMC University of Technology 13/10/200957:020 Fluid Mechanics
B.3. LIFT-Applications
Aspect ratio:
The effects of the finite aspect ratio can be characterized as a reduction Δα in the effective angle of attack
HCMC University of Technology 13/10/200957:020 Fluid Mechanics
B.3. LIFT-Applications
The drag of a wing of finite span:
HCMC University of Technology 13/10/200957:020 Fluid Mechanics
B.3. LIFT-Applications
It is possible to increase the effective AR for a wing of given geometric ration by adding an endplate or winglet to the wing tip.
Endplate (Airbus A320)
Winglet (Boeing 777)
HCMC University of Technology 13/10/200957:020 Fluid Mechanics
B.3. LIFT - Applications
Minimum Landing Speed
Two basic techniques:

1. Variable-geometry wing sections
HCMC University of Technology 13/10/200957:020 Fluid Mechanics
1. Variable-geometry wing sections
B.3. LIFT - Applications
HCMC University of Technology 13/10/200957:020 Fluid Mechanics
B.3. LIFT - Applications
2. Boundary-layer control techniques
(Delay separation or reduce drag, by adding momentum to B.L through blowing, or by removing low-momentum B.L fluid by suction)
Boeing 747
B.L control devices
HCMC University of Technology 13/10/200957:020 Fluid Mechanics
B.3. LIFT - Applications
2. Boundary-layer control techniques
(Delay separation or reduce drag, by adding momentum to B.L through blowing, or by removing low-momentum B.L fluid by suction)
HCMC University of Technology 13/10/200957:020 Fluid Mechanics
B.3. LIFT - Applications
2. Boundary-layer control techniques
(Delay separation or reduce drag, by adding momentum to B.L through blowing, or by removing low-momentum B.L fluid by suction)
HCMC University of Technology 13/10/200957:020 Fluid Mechanics
B.3. LIFT - Applications
2. Boundary-layer control techniques
(Delay separation or reduce drag, by adding momentum to B.L through blowing, or by removing low-momentum B.L fluid by suction)
HCMC University of Technology 13/10/200957:020 Fluid Mechanics
B.3. LIFT - Applications
2. Boundary-layer control techniques
(Delay separation or reduce drag, by adding momentum to B.L through blowing, or by removing low-momentum B.L fluid by suction)
HCMC University of Technology 13/10/200957:020 Fluid Mechanics
B.3. LIFT - Applications
2. Boundary-layer control techniques
(Delay separation or reduce drag, by adding momentum to B.L through blowing, or by removing low-momentum B.L fluid by suction)
HCMC University of Technology 13/10/200957:020 Fluid Mechanics
B.3. LIFT - Applications
2. Boundary-layer control techniques
(Delay separation or reduce drag, by adding momentum to B.L through blowing, or by removing low-momentum B.L fluid by suction)
HCMC University of Technology 13/10/200957:020 Fluid Mechanics
B.3. LIFT - Applications
2. Boundary-layer control techniques
(Delay separation or reduce drag, by adding momentum to B.L through blowing, or by removing low-momentum B.L fluid by suction)
HCMC University of Technology 13/10/200957:020 Fluid Mechanics
B.3. LIFT - Applications
2. Boundary-layer control techniques
(Delay separation or reduce drag, by adding momentum to B.L through blowing, or by removing low-momentum B.L fluid by suction)
HCMC University of Technology 13/10/200957:020 Fluid Mechanics
B.3. LIFT - Applications
Example 9.8: Optimum Cruise Performance of a Jet Transport
GIVEN: Boeing 727-200 jet transport at sea-level conditions
W = 150,000lbf, A = 1600ft2, AR = 6.5, CD,0 = 0.0182
Vstall = 175mph, compressibility effects on drag are negligible for M<0.6, sonic speed at sea level is c = 759mph
FIND: (a) Drag force as a function of speed from Vstall to M = 0.6; plot results
(b) Estimate of optimum cruise speed at sea level
(c) Stall speed and optimum cruise speed at 30,000ft altitude
Boeing 727
HCMC University of Technology 13/10/200957:020 Fluid Mechanics
B.3. LIFT - Applications
Example 9.8:
HCMC University of Technology 13/10/200957:020 Fluid Mechanics
B.3. LIFT - Applications
Example 9.8: Optimum Cruise Performance of a Jet Transport
HCMC University of Technology 13/10/200957:020 Fluid Mechanics
B.3. LIFT - Applications
Example 9.8: Optimum Cruise Performance of a Jet Transport
Boeing 727
HCMC University of Technology 13/10/200957:020 Fluid Mechanics
B.3. LIFT - Applications
Optimal Design of High-Speed Land Vehicles
At high speeds, aerodynamic lift forces can unload tires, causing serious reductions in steering control and reducing stability
Parasite Drag
HCMC University of Technology 13/10/200957:020 Fluid Mechanics
B.3. LIFT - Applications
Recreation (Golf, tennis, ping-pong, baseball…)
Players use spin to control the trajectory and bounce of a shot
HCMC University of Technology 13/10/200957:020 Fluid Mechanics
B.3. LIFT - Applications
Recreation (Golf, tennis, ping-pong, baseball…)
Boundary layer control
Conventional round dimples
Hexagonal dimples (Callaway HX)
HCMC University of Technology 13/10/200957:020 Fluid Mechanics
B.3. LIFT - Applications
Example 9.9: Lift of a Spinning Ball
GIVEN: A smooth tennis ball in flight, with m=57g and D=64mm, hit with V=25m/s and topspin of 7500rpm.
FIND: (a) Aerodynamic lift acting on ball
(b) Radius of curvature of path in vertical plane
(c) Comparison with radius for no spin
SOLUTION:
HCMC University of Technology 13/10/200957:020 Fluid Mechanics
B.3. LIFT - Applications
Example 9.9: Lift of a Spinning Ball