Fluid Mechanics II

Part 2I- Boundary Layer Theory

II- Potential Flow

Jafar GhazanfarianMechanical Engineering

Department

ghazanfarian.ir

2

Part I: External Flows OR

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.

ghaz

anfa

rian

.ir

3 new important phenomena in external flows are:

1- Stagnation pointNormal contact of a fluid particle to the body frontThe flow is divided into 2 branchesThe velocity is zero not due to the viscosity effectOn other points over body the velocity is zero due to no-

slip condition (roughness + viscosity) Not a viscous effect (It exists in potential flow)Normal component is zero: lack of porosity effectTangential velocity is zero: local symmetryBernoulli Equation holds

3

ghaz

anfa

rian

.ir

4

2- Boundary layer formationLudwig Prandtl (1904)Added a new horizon in fluid mechanicsThe 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

floorThe 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

roughnessNow, the boat is the fluid particle

ghaz

anfa

rian

.ir

5

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

◦ Flow region far from body in which: The flow is irrotational The Bernoulli Eq. holds Viscous effect is damped

Boundary Layer is a layer which is a boundary of two regions. As viscosity increases this region thickens or shrinks? 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

ghaz

anfa

rian

.ir

Re=VD/ ʋ

6

3- BL SeparationIn stagnation point V=0, P is maximumBefore reaching top:

◦ V increases, P decreases in flow direction

◦ Favorable pressure gradient

◦ Pressure and momentum acting in same directions

After passing the top point: ◦ V decreases, P increases in flow direction

◦ Strong adverse pressure gradient

◦ Pressure and momentum acting on opposite directions ghaz

anfa

rian

.ir

7

3- BL Separation 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? 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

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

ghaz

anfa

rian

.ir Re=VD/ ʋ

8

BL Flow regimes Similar to other viscous flows we have 3 flow regimes

◦ Laminar: over starting point of the body

◦ Transition: mid-locations of the body

◦ Turbulent: ending parts of the body

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)

ghaz

anfa

rian

.ir

9

Aero/hydro-dynamic forces in external flowsTotal force is the sum of two components:

◦ Parallel to the upstream flow direction drag force

◦ Normal to the upstream flow direction lift force

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

ghaz

anfa

rian

.ir

10

Aero/hydro-dynamic forces: goal in external flows 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

Assumptions of BL theory1- 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

ghaz

anfa

rian

.ir

11

Assumptions of BL theory4- High Reynolds number

◦ 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.

ghaz

anfa

rian

.ir

12

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).

Bernoulli‘s equation and BL 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.

ghaz

anfa

rian

.ir

13

Bernoulli ‘s equation and BL Bernoulli’s equation between BL edge and upstream flow (1):

Pressure coefficient is defined as:◦ P is pressure at the BL edge

◦ P0 is a reference pressure OR upstream pressure (1)

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

◦ Exact but Difficult

Integral approach: Von Karman integral◦ Approximate but Easy

ghaz

anfa

rian

.ir

Note

14

3 BL thicknesses1- BL Thickness

U=0.99 U∞

2- Displacement thickness:◦ 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: Momentum decreases relative to the inviscid flow due to viscosity effects

4- Energy thickness

Shape factor: δ>δ*>θ

ghaz

anfa

rian

.irNote

15

Viscous force computation on immersed bodies 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

Von Karman integral solution using control volume for arbitrary shapes

BL is not a streamline

There is an entrance of flow into the BL

ghaz

anfa

rian

.ir

Note

16

Von Karman (VK) Integral equation

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 integral1- 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

ghaz

anfa

rian

.ir

17

Solution procedure of VK integral Comment: The guessed velocity profile should satisfy the boundary

conditions:◦ U=0 @ y=0

◦ U=Us @ y= δ

◦ τ=0 @ y= δ

◦ d2U/dy2=0 @ y= δ

◦ …

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

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

ghaz

anfa

rian

.ir

Note

18

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:

◦ δ, θ, δ*, τw

◦ Cf: dimensionless wall shear stress

◦ FD,: integration of Cf over entire plate

Turbulent boundary layer

ghaz

anfa

rian

.ir

19

Turbulent boundary layer 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

. . .

The BL slope is increased relative to the laminar BL

All quantities are proportional to inverse of Re-1/5

ghaz

anfa

rian

.ir

Note

Example

20

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

Viscous drag coefficient computation 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

◦ The mixed drag force is less than the fully turbulent drag force

◦ In this case we use the modified Schilichting’s formula

ghaz

anfa

rian

.ir

NoteExample

21

Pressure drag (form drag) 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

ghaz

anfa

rian

.ir

Note

Example

22

Drag coefficient over thick bodies 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

ghaz

anfa

rian

.ir

23

Flow over cylinder and sphere Separation point varies with Re Transition occurs at 3.5×105

We introduce 3 new phenomenon:◦ Vortex shedding

◦ Vortex locked-on

◦ Von-Karman vortex street

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?

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

◦ Combined pressure and viscous drag force

ghaz

anfa

rian

.ir

24

Flow over cylinder and sphere 30<Re<104

◦ Vortex shedding starts

◦ Karman vortex street

◦ No 2D symmetry

◦ Oscillating drag and lift forces

◦ 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

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?

ghaz

anfa

rian

.ir

25

Flow over cylinder and sphere 3.5×105 <Re

◦ Transition to turbulent flow occurs

◦ Separation jumps from 80ᵒ to 115ᵒ. Why?

◦ CD decreases up to 50 %! Very nice!

◦ Golf ball: artificial transition to turbulent flow by wall roughness to use the benefit of 50 % decrease

◦ Then CD starts to increases

ghaz

anfa

rian

.ir

26

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

ghaz

anfa

rian

.ir

2 Examples

27

Flow over bluff bodies Roughness effects

ghaz

anfa

rian

.ir

28

BL control OR drag reduction Thin bodies airfoil, flat plate

◦ No separation

◦ Smoothed surface

◦ Keep flow laminar

Bluff bodies◦ Use 50 % reduction for turbulent flow by roughness

◦ Streamlining, aerodynamic design

◦ Controls flow separation

ghaz

anfa

rian

.ir

29

BL control OR drag reductionOther methods

◦ Suction

◦ Injection (reenergizing)

◦ Slot Low mechanical design

◦ Wall movement Wall rotation

ghaz

anfa

rian

.ir

30

Attack/incident angle Very important parameter for control of separation Angle between upstream flow and the body (f35) High attack angle leads to Stall

ghaz

anfa

rian

.ir

31

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

Aspect ratio is defined as the ratio of the square of the wing length to the planform area=b2/A=b2/bc=b/c

ghaz

anfa

rian

.ir

32

Polar curve Polar curve is the variation CL against CD

Finite Wing 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

ghaz

anfa

rian

.ir

Finite Wing Finite wing has less lift force, greater drag force, smaller CL/CD

Cruise flight 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

33

ghaz

anfa

rian

.ir

Note

34

Flaps Fowler: airbus 340/330+ Boeing 777 There are other kinds: leading edge, double slotted

ghaz

anfa

rian

.ir

35

Rotating sphere/cylinder (CW) 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 bott

om moves towards upstream Streamlines are denser on top Velocity increases Bernoulli says

the pressure on top is reduced An upward lift force is created

ghaz

anfa

rian

.ir

Re=105Example

36

Differential equations of BL

Blasius Solution Flow over flat plate Write full N.S. equations + continuity

Use Prandtl BL assumptions

Perform order of magnitude analysis to omit some terms

ghaz

anfa

rian

.ir

Note

37

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)

ghaz

anfa

rian

.ir

Note

Example

38

Types of Drag force1- 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

ghaz

anfa

rian

.ir

39

Introduction to Airfoils 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

ghaz

anfa

rian

.ir

2 Examples

40

Part II:Potential flow

ORIdeal flow

OROut-of-BL flow

ORInviscid flow

ORNon-Rotational Flow

The flow in which the effect of viscosity can be ignored.

ghaz

anfa

rian

.ir