Transcript Fluid Mechanics II

Fluid Mechanics II
Part 2
I- Boundary Layer Theory
II- Potential Flow
Jafar Ghazanfarian
Mechanical Engineering
Department
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Part I: External Flows
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Boundary Layer Theory (BL)
External flow: Flow over bluff/blunt bodies which are
surrounded by an infinite fluid.
 External flow is a special case of internal flow.
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3 new important phenomena in external flows are:
1- Stagnation point
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Normal contact of a fluid particle to the body front
The flow is divided into 2 branches
The velocity is zero not due to the viscosity effect
On other points over body the velocity is zero due to noslip condition (roughness + viscosity)
Not a viscous effect (It exists in potential flow)
Normal component is zero: lack of porosity effect
Tangential velocity is zero: local symmetry
Bernoulli Equation holds
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2- Boundary layer formation
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Ludwig Prandtl (1904)
Added a new horizon in fluid mechanics
The first semi-analytical solution of the N.S.
A 3-unit course in MSc, with a 800-page reference book
An example of a boat moving over a outgrowth on the
floor
The distance through which the effect of outgrowth is
sensed by you?
How this distance changes by travelling in flow direction?
Combine millions of these outgrowths to form the surface
roughness
Now, the boat is the fluid particle
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2- Boundary layer formation
 We divide the flow region around bodies into 2 zones:
◦ The near-wall region in which:
 The flow is rotational
 The Bernoulli’s Equation fails
 Viscous effect is important
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◦ Flow region far from body in which:
 The flow is irrotational
 The Bernoulli Eq. holds
 Viscous effect is damped
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Boundary Layer is a layer which is a boundary of two regions.
As viscosity increases this region thickens or shrinks?
Re=VD/ʋ
As flow velocity increases this region thickens or shrinks?
As flow regime changes to turbulent this region thickens or shrinks
(in constant Re) ?
Starts from stagnation point
Along BL the velocity profile is drawn OR it’s slop is reduced
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3- BL Separation
In stagnation point V=0, P is maximum
 Before reaching top:
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◦ V increases, P decreases in flow direction
◦ Favorable pressure gradient
◦ Pressure and momentum acting in same directions
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After passing the top point:
◦ V decreases, P increases in flow direction
◦ Strong adverse pressure gradient
◦ Pressure and momentum acting on opposite directions
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3- BL Separation
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Adverse pressure gradient is necessary condition for separation
but not enough
Pressure gradient is a geometric parameter
Separation should be prevented
As flow velocity increases
what happens for separation?
Re=VD/ʋ
As flow regime changes to turbulent what happens for
separation (in constant Re)?
Turning-point for
◦ Separated flow is located in the flow
◦ In the separation point is located on the body
◦ Before separation point is located inside the solid body
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The shear stress on the wall in the separation point is zero.
There is a separate flow/reversed flow/back flow with low
pressure region
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BL Flow regimes
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Similar to other viscous flows we have 3 flow regimes
◦ Laminar: over starting point of the body
◦ Transition: mid-locations of the body
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◦ Turbulent: ending parts of the body
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Re = xV/ʋ is geometric parameter
◦ X is a coordinate placed over body and starting from the stagnation point
◦ The BL thickness grows faster in turbulent flow (higher BL slope)
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Aero/hydro-dynamic forces in external flows
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Total force is the sum of two components:
◦ Parallel to the upstream flow direction drag force
◦ Normal to the upstream flow direction  lift force
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The two origins of total force are:
◦ Pressure force: normal to surface
◦ Shear force: tangent to surface
◦ In a motion:
 Thrust  Drag force
 Weight  Lift force
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Aero/hydro-dynamic forces: goal in external flows
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Using dimensional analysis:
We can find the f using experimental or analytical or numerical
methods. In BL theory we are going to use analytical approach.
 The separation controls the pressure lift/drag
 Viscosity and the wetted area control the shear lift/drag
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Assumptions of BL theory
1- incompressible flow
2- steady state
3- 2D flow tangent and normal to the body
◦ x is attached to the body surface (curvilinear coordinate)
◦ The curvature of the surface cannot be sensed by the fluid particle
◦ High radius of curvature (like smooth earth and humans)
4- High Reynolds number
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Assumptions of BL theory
4- High Reynolds number
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Re=Ux/ʋ >105-107
The viscous force is small relative to the inertia force
It seems to be a paradox but it is not.
Flow can be laminar or turbulent.
The viscous force should be small for the use of BL theory
WHY?
◦ NEVER FORGET: THE VISCOUS FORCE NEVER CAN BE
NEGLECTED.
‫فلفل نبین چه ریزه بشکن ببین چه تیزه‬
◦ When the viscous force is small the region which is affected by
the viscous force is small.
◦ So, the BL is very thin.
◦ So, to use the BL theory the thickness of BL should be small.
BUT NOT ZERO
◦ Thinner boundary layers are more destroying.
◦ Great amount of heat is generated in boundary layer.
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D’alembert Paradox
The drag force of an inviscid flow over any arbitrary shaped object
is zero, which is obviously a wrong statement.
 This paradox was solved when the viscous terms were added to the
Euler equation (1755) to create the full NS equations (1855).
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Bernoulli‘s equation and BL
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Why the BL should be assumed to be thin?
◦ The pressure distribution at the edge of BL is a known parameter in BL theory.
◦ We want to estimate the pressure at the edge of BL by the pressure on the body.
◦ We can compute the pressure over the body using Bernoulli’s equation
◦ We neglect the pressure gradient normal to the BL.
◦ This fact is valid when the BL is thin
◦ Example of a paper and a cleaner over the whit board.
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Bernoulli ‘s equation and BL
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Bernoulli’s equation between BL edge and upstream flow (1):
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Pressure coefficient is defined as:
◦ P is pressure at the BL edge
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◦ P0 is a reference pressure OR upstream pressure (1)
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This parameter is 1 at the stagnation point and then starts to
decrease
2 Students of Prandtl
We learn two approaches in this course:
 Differential approach: Blasius solution
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◦ Exact but Difficult
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Integral approach: Von Karman integral
◦ Approximate but Easy
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Note
3 BL thicknesses
1- BL Thickness
U=0.99 U∞
2- Displacement thickness:
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◦ The flow decreases relative to the inviscid flow due to viscosity effects
◦ Displacement of streamlines relative to the inviscid flow due to
viscosity effects
3- Momentum thickness:
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Momentum decreases relative to the inviscid flow due to viscosity effects
4- Energy thickness
Shape factor:
δ>δ*>θ
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Viscous force computation on immersed bodies
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Viscous force is created due to the shear stress on wetted surfaces
In order to ignore the pressure force we ignore the separation
The pressure gradient is favorable. Pressure force is zero
If the plate is thick, the pressure force is created (square)
The flat plate is the most important case
Note
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Von Karman integral solution using control volume for arbitrary
shapes
BL is not a streamline
There is an entrance of flow into the BL
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Von Karman (VK) Integral equation
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U is the velocity of BL edge known from the Bernoulli’s equation
The first term is surface force originated from shear stress
The second term is the exchange of momentum from C.S.
The last term is surface force originated from pressure
Is valid for both laminar and turbulent flows
A combined integral-differential equation difficult to solve
Solution procedure of VK integral
1- Obtain U from Bernoulli’s equation out of BL and potential flow
2- Guess the velocity profile within the BL
3- Compute the shear stress (Newtonian fluid) , displacement and momentum
thicknesses
4- Solve the final ODE to fined δ
5- Substitute computed δ to find the shear stress on the wall
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Solution procedure of VK integral
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Comment: The guessed velocity profile should satisfy the
boundary conditions:
◦ U=0 @ y=0
◦ U=Us @ y= δ
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◦ τ=0 @ y= δ
◦ d2U/dy2=0 @ y= δ
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The second order polynomial is a good profile but with non-zero
second order derivative.
 This profile is good when there is no separation Flat Plate
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VK integral solution for flat plate
U=cte
 Second order polynomial and sinusoidal profile are selected
 δ, θ, δ*, τw in laminar BL of flat plate is proportional to inverse
of square root of Rex
 The drag force is totally viscous force no pressure drag
Compare with the exact Blasius solution
Consider the geometrical and physical interpretation
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VK integral solution for flat plate
The drag coefficient is proportional to the momentum thickness at
the end of plate
 There is no separation (dP/dx=0)
 We computed:
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◦ δ, θ, δ*, τw
◦ Cf: dimensionless wall shear stress
◦ FD,: integration of Cf over entire plate
Turbulent boundary layer
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Turbulent boundary layer
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Similar idea stemming from the internal flow solution
In internal flow BL edges make contact
The BL edge pressure is not constant anymore
In external flow there is no BL contact!
The BL edge pressure is constant (for flat plate)
This approximation is valid for 5×105< Re<107
We use the power-law velocity profile to compute δ, θ, δ*
But τw is computed using Blasius formula for smooth pipe
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Example
The BL slope is increased relative to the laminar BL
All quantities are proportional to inverse of Re-1/5
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Pressure gradient in BL
In order to obtain zero shear stress at wall, two terms on RHS must
be with opposite signs
 Separation occurs in the positive pressure gradient
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Viscous drag coefficient computation
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The pressure force is supposed to be zero
Drag force (FD) is the integration of wall shear stress (τW)
Drag coefficient (CD) is the integration of friction coefficient (Cf)
These formulas are valid for 5×105< Re<107 . Why?
For 107< Re<109 we use the Schilichting’s formula
If the flow is initially laminar and then becomes turbulent
Example
◦ The mixed drag force is less than the fully turbulent drag force
◦ In this case we use the modified Schilichting’s formula
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Pressure drag (form drag)
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The geometry is in a way that the drag force is totally pressure drag
Similar to totally viscous case the body is thin
The pressure distribution around body (CP) should be known
In such cases the separation point is not a function of Re
Example for a vertical flat plate
Pay attention to the sign of integrals
Example
Note
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Drag coefficient over thick bodies
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We have studied two limiting cases of horizontal and vertical flat
plates
Now we study thick bodies
The drag force is mixed pressure-viscous force
The separation point is dependent on Re and roughness
The separation controls the portion of pressure drag
Wetted area and viscosity control the portion of viscous drag
The separation creates a relatively constant low pressure region
behind body
If flow separates earlier the pressure drag increases
Flow over ellipse
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Flow over cylinder and sphere
Separation point varies with Re
 Transition occurs at 3.5×105
 We introduce 3 new phenomenon:
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◦ Vortex shedding
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◦ Vortex locked-on
◦ Von-Karman vortex street
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Re<1: Low Re flow/ creeping flow/ Stokes flow
◦ Drag force is totally viscous
◦ No data for cylinder!!!
◦ Steady flow: L=0; D≠0; Why?
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1<Re<30
◦ Flow separates near back stagnation point
◦ Separation point moves towards the upstream. Why?
◦ Steady flow: L=0; D≠0; Why?
◦ Symmetric and attached vortices
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Combined pressure and viscous drag force
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Flow over cylinder and sphere
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30<Re<104
◦ Vortex shedding starts
◦ Karman vortex street
◦ No 2D symmetry
◦ Oscillating drag and lift forces
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◦ L (t)≠ 0; D(t)≠0; Why?
◦ Lmean=0; Dmean ≠0
◦ Flow induced Vibration of cylinder
◦ Dimensionless Frequency of oscillation is called Strouhal number
◦ St=f D/V=0.2 ( 1 - 20/Re ) ≈ 0.2 (high Re) ≡ 5 second of one period
◦ Resonance (Vortex Locked-on): equality of f with natural frequency of
structure
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104<Re< 3.5×105
Re= 104
◦ Still laminar
◦ Vortex shedding is intense
CD is slightle increase (constant) Why?
D’alembert paradox
Separation is at 80ᵒ? Why?
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Flow over cylinder and sphere
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3.5×105 <Re
◦ Transition to turbulent flow occurs
◦ Separation jumps from 80ᵒ to 115ᵒ. Why?
◦ CD decreases up to 50 %! Very nice!
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◦ Golf ball: artificial transition to turbulent flow by wall roughness to use the
benefit of 50 % decrease
◦ Then CD starts to increases
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Flow over cylinder and sphere
Pressure distribution: sphere/ cylinder
 Lower pressure in separated flow for
laminar regime
 CD versus Re
 Make the flow turbulent to decrease the pressure behind the body
 Other bluff bodies
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2 Examples
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Flow over bluff bodies
Roughness effects
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BL control OR drag reduction
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Thin bodies airfoil, flat plate
◦ No separation
◦ Smoothed surface
◦ Keep flow laminar
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Bluff bodies
◦ Use 50 % reduction for turbulent flow by roughness
◦ Streamlining, aerodynamic design
◦ Controls flow separation
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BL control OR drag reduction
 Other methods
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◦ Suction
◦ Injection (reenergizing)
◦ Slot
 Low mechanical design
◦ Wall movement
 Wall rotation
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Attack/incident angle
Very important parameter for control of separation
 Angle between upstream flow and the body (f35)
 High attack angle leads to Stall
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Stall
Sudden reduction of lift force for any reason
 This reason may be separation for high attack angle
 There is an attack angle for maximum lift coefficient
 For short landing distances sometimes stall is very helpful
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Aspect ratio is defined as the ratio of the square of the wing length
to the planform area=b2/A=b2/bc=b/c
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Polar curve
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Polar curve is the variation CL against CD
Finite Wing
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For infinite wing the aspect ratio is infinity
Relating high and low pressures down and top of the wing creates a
secondary flow called trailing vortex (TV)
This vortex reduces the net lift force
More heavy the aircraft more powerful these vortices
These vortices creates two line of water vapor in the sky
These vortices may exist up to 10 miles (15 Km) away from the
airplane path and several hours after flight
Very dangerous for small airplanes
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Finite Wing
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Finite wing has less lift force, greater drag force, smaller CL/CD
Cruise flight
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Note
V2=2W/(CLAρ)
 V must be minimized but greater than 1.1Vstall
 W should be reduced, CL,A,ρ should be increased
 Flaps help us increase A
 Flaps also increase CD, So they are used in landing
 The cross section of wing increases by reaching the body, Why?
Airbus A319
A340
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Flaps
Fowler: airbus 340/330+ Boeing 777
 There are other kinds: leading edge, double slotted
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Rotating sphere/cylinder (CW)
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Nonsymmetrical flow which creates lift force
Momentum increases on top and decreases underneath
Stagnation point moves towards the bottom side and then enters the
fluid
The separation point on top moves towards downstream and on
bottom moves towards upstream
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Re=10
Streamlines are
denser
on top Velocity increases Bernoulli
says the pressure on top is reduced An upward lift force is
created
Example
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Differential equations of BL
Blasius Solution
Flow over flat plate
 Write full N.S. equations + continuity
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Note
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Use Prandtl BL assumptions
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Perform order of magnitude analysis to omit some terms
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Differential equations of BL
Blasius Solution
Omit velocity from equations using stream function concept
 Convert Prandtl equation (PDE) to Blasius equation (ODE) by:
Similarity solution
 Defining similarity parameter
 Obtain f (flow rate), f’ (velocity), f” (shear stress)
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Example
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Types of Drag force
1- Skin friction drag shear stress
2- Pressure drag (form drag) separation
3- Profile drag (1+2)
4- Sink drag (energy needed for injection and suction)
5- Wave making drag (free surface flows on floating body) bulb
6- Wave drag (free surface on immersed body)
7- Induced drag
8- compressible wave drag
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Introduction to Airfoils
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Trailing/ leading edges
Symmetrical wings need more attack angle, CL=0
Chord length (c)
Camber line (h: distance to chord)
Span length (b)
A=bc
Thikness: t
Re is defined based on chord length
NACA 2415 For nonsymmetrical airfoils
hmax=0.2c, b=hmax is placed at 0.4c, cd=tmax=0.15c
NACA 0012 For symmetrical airfoils
2 Examples
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Part II:
Potential flow
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Ideal flow
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Out-of-BL flow
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Inviscid flow
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Non-Rotational Flow
The flow in which the effect of viscosity can be ignored.
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