Spark Notes - Flow Past a Body

7/23/2019 Spark Notes - Flow Past a Body

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Flow past a Body:

Passing flow exerts pressure forces and viscous forces on a body

The resolved components of these forces produce dragand lift.

The sizes of these forces are affected by the size and shape of the body and the flow

Reynolds number

Figure 1: showing the action of the pressure and viscous forces on an aerofoil

Dragacts parallel to flow velocity

Drag Coefficient: = The flow structures and associated forces of various objects are functions of both shape and Re and

therefore: the objects shape, its orientation to the flow and Re determine the drag coefficient.

Liftacts perpendicular to flow velocity

Lift Coefficient: = *Remember (F= the drag (D) and Lift (L) forces: unit N S = representative area for body: unit m2)

Representative area for certain forms:

Shape Flow diagramRepresentative

area

Sphere

Characteristic length = d

S =d2/4

Finite Cylinder

Characteristic length = d

S = bd


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Aerofoil

c= Chord b= SpanCharacteristic length = c

S = cb

The Total drag(or profile drag) on a body incorporates two different components of drag:

TOTAL DRAG (PROFILE DRAG)= FORM DRAG + SKIN FRICTION

Form Drag: Resultant forcesacting normal to the surface due to pressure

distribution/normal stresses. Pressure (p)acts on the surface area of the body (ds) resulting

in the form drag:

FORM DRAG = Skin Friction: The shear force that acts tangential to the surface (along the stream) due to

velocity gradient normal to the surface. Shear stresses (w) act on the surface area of the

body (ds)resulting in the shear force:

SKIN FRICTION = Subcritical Re:Laminar separation before the maximum transverse dimension (vertical maximum

looking at cross section)a wide wake at low pressure Large form drag and negligible skin friction

Supercritical Re: Turbulent separation after maximum transverse dimension narrow wake at

higher pressure low form drag and enlarger but still negligible skin friction.

The Pressure Coefficient:

The flow around the body accelerates and decelerates the local static pressure therefore is

different from that of the main stream

This static pressure can be found using Bernoullisoutside of the very thin boundary layerdue

to its negligible thickness this pressure is transmitted through the un-separated boundary layer

onto the surface.

*RememberBernoullis + = +

For different shapes or forms of body the pressure distribution is described by the pressure

coefficient -

Pressure Coefficient: =

= 1


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Figure 2: Cpvs angle plots for inviscid theory and experimental data

Inviscid and viscous flow past a body:

Looking at Reynolds number; a high Reynolds number can suggest that viscous forces are

insignificant versus inertia forces but this is NOT the case

Inviscid theory Experiment at high Re

For Flow past a cylinder:

Predicts no drag ( = 0) Large drag observed : high form drag, negligible skinfriction CD 1 at high Re

For Flow past an Aerofoil:

Predicts no Lift ( = 0)Predicts no Drag ( = 0)

OR


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Can Predict Lift ( = 0)

Predicts no Drag ( = 0) Predicts no Drag ( = 0): dominant skin friction, lowform drag CD 0.01 at high Re

*Remember (Inviscid Theory = assuming zero viscosity)

Potential Flow: flow of an incompressible fluid that is assumed to be frictionless and motion to be

irrotational.

The conditions for potential flow are:

Ideal Fluid

o

No viscosity, no compressibility, no surface tension, no vaporisation

Irrotational

o No rotation, no vorticity, no circulation

Flow Separation: caused by a reduction of velocity in the boundary layer and a positive

(adverse) pressure gradient. Flow separation leads to a large form drag.

Prandtls boundary layer concept (see topic 1)in high Reynolds flow = Viscous forces are small but

NOT negligible

o

The Outer flow : behaves as inviscid - velocity gradients are very smallo The Boundary layer: where viscous effects are limited to a thin layer at the objects/walls

surface (shear stress exists at the wall due to velocity gradients at the surface. The boundary

layer can be considered to have a high momentum deficit.

N:BThe thickness of the boundary layer is very small at high Re, no slip condition applies.

Figure 3: Showing the process of flow separation


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Before the separation point:

Outer Flow is accelerating:U(x) is increasing and p(x) is decreasing

There is a favourable pressure gradient >

< (negative)

At and after the separation point:

Outer Flow is Decelerating: U(x) decreasing and p(x) increasing

There is an adverse pressure gradient < > (positive)

The adverse pressure gradient slows further the already slowed flow close to the wall

The flow at the wall comes to rest, then it reverses and separates.

N:B Separation occurs around the point where = 0and = 0and is caused by

Laminar and Turbulent Flow separation:

Figure 4: Laminar flow separation, left. Turbulent Flow separation, right

Laminarflow is unable to adjust to the increasing pressure and flow separation occurs

Turbulentboundary layers are more resistant to separationo The chaotic nature of the turbulent boundary layer allows for mixing and absorption

of more energy from the free stream

o The increased mixing increases the transport of momentum (due to turbulent

stresses between the free-stream and the flow near the wall.)

o This transport of momentum from the free-stream to the wall increases stream-wise

momentum in the boundary layer.

o This allows the turbulent boundary layer to overcome the pressure gradient and

resist separation for longer.

Boundary layer controlprevention of flow separation: these systems can be expensive and hard to

install, all control methods are specific to situation and condition (advantages maybe be offset by

additional drag)

(1) Surface rougheninginitiates turbulence in the boundary layerdelays separation and

produces a narrower wake of higher pressure

(2) Shape Variationincorporating a flap at the end of a wing and tilting before separation

occursincreasing the top side (of an aerofoil) velocity, reducing the pressure and thus

chance of separation.


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(3) Fluid injectionaccelerates the flow of the boundary layer (but increases turbulence of flow

and thus skin friction)

(4) Fluid removalSuction of fluid through small holes or porous surfacethins and speeds up

the downstream boundary layer increasing resistance to an adverse pressure gradient

laminar to turbulent transition is delayed leading to less skin friction

(5) Slotted wingslots in the wing draw air from the higher pressure base area of the wing up

to the topthe slow moving air on the upper surface is accelerated, decreasing pressure

and reducing the adverse pressure gradient.

(6) Engine arrangementengine intakes draw slow air from the boundary layer at the rear of

the wing through small holesthe boundary layer remains close to the wing and this

produces higher pressure gradients before separation.

Hydrofoils:

- Most Hydrofoils are used limit the onset of cavitation and boundary layer separation.

- Used to lift vessels out of water and also on control surfaces, rudders and stabilisers.

- The most important piece of information of hydrofoil design is the location of the minimum

pressure (where cavitation will first occur)

Cavitation: Occurs when the pressure in flow drops local below the vaporisation pressure of the

fluidlocal boiling (a phase change) of the fluid occurs. The cavitation number is =

Streamlining: lowering the form drag by reducing the tendency of the flow to separate, which

essentially lowers the coefficient of drag.

Flow past a cylinder:

Flow separation

Flow accelerates as it moves around the cylinder, reaching a maximum at the vertical axis of

the circular cross section. The pressure has dropped and thus the adverse pressure gradient

created causes flow separation as long as flow is fast enough (Re>2)

At flows of Re 30 - 70 steady flow exists and above this point oscillations start to appear

At flows of Re > 70 90 the separation point moves upstream and the separated zones curl

into vortices which detach and trail on the downstream side of the cylinder: this is known as

a Krmn vortex trail.

This vortex shedding can induce aerodynamic flutter which can have large effects on the

design of structural components

Vortex shedding:

Frequency of vortex shedding is given in terms of the Strouhalnumber = (Where n= frequency of vortex shedding (Hz), = the characteristic length of the object (m) and=free stream velocity ( ))


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Drag

-

For a cylinder CD is nearly constant for Re = 103-105 but varies as the boundary layer is

turbulent between Re = 105-106

- At Re 2000: CD is at a minimum of approximately 0.9. At Re 30,000: CD rises toapproximately 1.2 (partially due to increasing turbulence in the wake). At Re 200,000: CDdrops suddenly to approximately 0.3 and for Re>200,000 CD increases slowly.

-

Form drag is approximately 50% of the profile drag as Re tends to 0. By Re 200, the formdrag is approximately 90% of the profile drag. At Re 30,000 skin friction is insignificant.

- Drag crisis : Recrit 2 1 05For a cylinder: The Pressure coefficient can be related to both the drag and lift coefficients

by:

=

=

Flowpast a sphere:

Seperation

-

Flow starts to separate from the spheres surface Re 20- As Re increases a pair of recirculating vortices become visible in the wake region-

At flow up to Re 200 the flow is steady and above this value oscillations start to appearand the separation point moves upstream of the top of the sphere which increases the drag.

Drag

-

For a sphere CDis nearly constant for Re = 103-105 but varies as the boundary layer becomes

turbulent between Re = 105-106 (this transition causes CD to decrease dramatically)

- As Re tends to 0 , CDis given by Stokes law CD=24/Re

- Drag crisis: Recrit 3 1 05Drag Reduction: Boundary layer tripping

- Boundary layer tripping is when a trip wire is deliberately incorporated to turn the boundary

layer from laminar to turbulent this increases skin friction but decreases form drag by

delaying flow separation

-

The Reduction in form drag > The Increase in skin friction and the overall CDdecreases.

- Another way to reduce drag is roughening the surface which lowers Re at which the

transition occurs but CD will rise faster once the transition point is passed.


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Flow past an aerofoil:

Figure 5 : A geometrically labelled aerofoil

Figure 6: Cp versus position on chord for upper and lowers surfaces of the aerofoil (left) and pressure

pattern on the upper surface (right)

- Velocity increases as air flows over the aerofoil and therefore pressure reduces above the

aerofoil inducing a transverse lift force.

-

High Re and low angle of attack a CLand CDdepend on a but are relatively insensitive to Re

Angle of Attack

- The angle of attack is the angle the chord line makes with the approach of fluid flow

-

For a small angle of attack separation is close to the trailing edge the wake is thin and

there is low form drag

- As angle of attack increases separation moves along the top of the aerofoil increasing the

wake width and form drag.

-

If the angle of attack is too large the boundary layer at the top of the aerofoil separates

(where adverse pressure gradients are large), increases form drag considerably and changes

the pattern of pressure this reduces the lift component to a value lower than the weight of

the aerofoil (the plane drops) known as stalling.

- The critical angle of attack is acritat which CLis a maximum, above which stalling occurs.

- The critical angle depends on the type and shape of the aerofoil.

Efficiencythe minimum CDdoes not occur at the maximum CLand therefore the ratio CL/CD can be

taken as a measure of the efficiency of an aerofoil.


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Bibliography

[1] A. Bakker, "Lecture 11 : Boundary Layer and Separation," 2002-2006. [Online]. Available:

http://www.bakker.org. [Accessed 04 9 14].

[2] D. P. Aleiferis, Mechanics of Fluids Lecture Notes, London, 2013-2014.

[3] P. J. A. T. Robert W.Fox, Introduction to Fluid Mechanics (seventh edition).

[4] B. M. (. b. J. Ward-Smith), Mechanics of Fluids (8th edition).

Spark Notes - Flow Past a Body

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