THERMOFLUIDS, Level5

Lecture 9 – Fluids (Internal & External Flows)

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-Turbulence origins

-Boundary layer

-External flows; Viscous & Turbulent, Drag & Lift coefficient, Stokes Law

-Internal flows; Viscous & Turbulent, Velocity profile, Velocity pressure, Blasius equation,

Friction factors (f), Mean & max velocity, Darcy-Weisbach equation, Moody Chart

Turbulence origins

2

Turbulent flow occurs when inertial forces dominate fluid flow, as opposed to viscous forces which dominate laminar

flow.

Turbulent flow – unsteady, disordered. Can be characterised by turbulent length scales.

Laminar flow – highly ordered smooth layers.

Bulk velocity may be same for

laminar & turbulent flow, but

individual particle velocity

unsteady. Greater energy transfer

within fluid, and between fluid and

boundary.

Boundary layer

3

•Boundary layer flow, shear stress and viscosity determine

skin friction.

•In both laminar and turbulent flows, the velocity is not uniform

but varies from zero at the surface to a maximum some

distance away.

•Velocity distribution is dependent upon the Reynolds number

which defines type of flow.

External flows – Viscous flow

4

Two distinct geometrical cases exist for fluids flows, in addition, fluids may flow in 2 regimes; laminar flow (or

viscous flow) and turbulent flow.

External flow – where a body (eg automotive vehicle, ship, aircraft) moves relative to a body of fluid.

Boundaries are infinite.

Internal flow – where fluid moves relative to external geometry (eg pipe or duct).

Stokes Law

(creeping flow)

D=2r V (m/s) Fluid dynamic viscosity

μ (kg/ms)

Using dimensional analysis

(density omitted as for creeping flow, inertial forces negligible),

Force F=const.raμbvc

The constant of proportionality has been shown to be equal to 6π, hence:

Fsphere= 6πrμv

Stokes Law (valid for very low Re, ‘creeping flow’)

External flows – Viscous flow – Cont…

5

A sphere has projected area πr2 and the fluid has relative velocity pressure of (ρv2/2). If the drag force is

divided by the projected area to become pressure, and then divided by the relative velocity pressure, a new

dimensionless group is formed:

DIA

D

sphere

vrvr

vC

vr

F

Re

2412

2/

6

)2/)(( 222

Where CD is the drag coefficient

Holds for ReDIA <0.1-2.0 (limit for external viscous flow)

Application – particles falling under gravity, at terminal

velocity where force due to gravity = drag. Terminal

velocity given by:

18

)(2

min

gDv

fluidsolid

alter

(valid for

sphere only,

creeping flow)

External flows – Turbulent flow

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As flow velocity increases, separation occurs and turbulence commenses in whole or in part (diagrams below

relate to simplified case of 2D cylinder – infinitely long).

Re<0.5

103<Re<2x105

Re>2x105

When separation takes place,

streamlines do not recover fully

downstream, nor does the static

pressure. The resulting ∆P gives

another component of drag – form or

profile drag, in addition to surface or

skin drag.

Form drag is dominates at high Re.

At Re>100,000 the boundary layer itself

becomes turbulent, which has the effect

of delaying stagnation and separation,

paradoxically reducing drag.

External flows – Turbulent flow – Cont…

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Strong differences in boundary layer

separation on a 8.5inch ball entering water at

25ft/sec. (a) smooth ball, laminar boundary

layer, (b) turbulent boundary flow induced by

patch of nose roughness.

Boundary

layer transition

DIA

DCRe

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External flows – Drag & Lift

8

Drag Coefficient:

Although CFD used increasingly for computation of forces acting on complicated geometries,

traditionally this has been achieved through experimental determination of drag coefficients.

In general, drag coefficient defined as:

AU

FC D

D2

2

1

CD=drag coefficient

FD=Force

ρ=density

U=free-stream velocity

A=projected area

2D shapes – the cylinder (A) and the airfoil (B) have the same frontal area, but the drag on the

cylinder is much greater.

External flows – Drag & Lift – Cont…

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Lift Coefficient:

Defined in a similar way to drag coefficient, but reference area is normally plan area (eg aircraft

wing chord x span):

AU

FC L

L2

2

1

CL=lift coefficient

FL=Lift force

ρ=density

U=free-stream velocity

A=projected area

Effect of ground proximity on the

aerodynamic lift and drag of an

ellipsoid

(width/height=1.25,length/height=

3.6,max thickness is at 1/3length)

External flows – Drag & Lift – Cont…

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Internal flows – Viscous flows

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Examples – airflow in ducts, water flow in pipes

Vmean (m/s)

Fluid dynamic viscosity

μ (kg/ms)

r=D

/2 (

m)

Q (m3/s)

Length, L (m)

Pressure drop, ∆P (Pa)

Variables in system:

Volume flow-rate, Q

Radius of duct, r

Dynamic viscosity, μ

Length, L

Pressure drop, ∆P

Mean velocity, v, and diameter D are not independent

variables as they are calculable from Q,r.

Internal flows – Viscous flows – Cont…

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Dimensions:

[Q]=L3T-1

[μ]=ML-1T-1

[r]=L

[∆P/L]=ML-2T-2

Using dimensional analysis:

Q=k μarb (∆P/L)c

L3T-1= (ML-1T-1) a(L)b (ML-2T-2)c

→a=-1, b=4, c=1.

The constant of proportionality has been

shown =π/8.

L

PrQ

8

4

Poiseuille’s equation (valid for low speed

viscous/Laminar flow only, where viscosity not

turbulence is the mechanism for resistance to

flow)

It may also be expressed:

4 as,

32 22

2

vDvrQ

D

LvP

The pressure-drop ∆P, is related to the shear-

stress at the wall τwall. The force balance is:

2

2 32

D

4 thus,

4 D

LvLDL

DP wallwall

Dividing both sides by velocity pressure, the

reference for moving fluid:

∆P x Area = ShearStress x Area

222 )2/v(

32

)2/v(

4

D

Lv

D

Lwall

The term

2/v2

wall is known as the fanning friction factor f

Re

1616

vDf

(Valid for viscous,

laminar flow)

Internal flows – Viscous flows - Velocity profile

13

meany

y

y

y

uu

r

L

Pu

and

r

y

u

u

2

4

)(1

0

2

0

2

2

0

For laminar flow, the velocity profile inside a tube of diameter

2r, distance y from centre-line is:

Internal flows – Turbulent flows – Velocity profile

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Poiseuille’s analysis breaks down as laminar flow ceases to apply. Dimensional analysis is more complex but

states friction factor(f) is a function of Reynolds number(Re) and wall roughness ratio(k/D).

A

Qu

uu

and

r

y

u

u

mean

meany

y

y

817.0

)(1

0

7

1

0

For turbulent flow, the velocity profile inside a duct, diameter D, distance y from the

centre-line is:

Internal flows – Turbulent flows – Dimensional analysis

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Determining friction factor(f)

Variables in system:

Density, ρ

Velocity, u

Diameter, D

Dynamic viscosity, μ

Roughness, k

Wall shear stress, τ

Dimensions: [L], [M], [T]

According to Buckingham,s law, must be 6-3=3

dimensionless groups. These groups are:

Relative roughness = k/D

f/2 = τ/(ρu2)

Reynolds number, Re= ρuD/ μ

The relationship between these groups is complex.

Various engineers have presented solutions applicable to

different ranges of fluid flow (Re).

•Colebrook and White give, for Re>2100

f-0.5=-4log10(0.27k/d+1.255(Re)-1f-0.5)

This equation is not explicit and has to be solved

iteratively.

•Moody gives, approximately for 4000<Re<10,000,000,

k/d<0.01

f=0.001375(1+(20,000(k/d)+1,000,000/Re)0.33)

•Blasius gives, for k/d=0 (smooth pipes) and

4,000<Re<100,000

f=0.079Re-0.25

•Poiseuille gives, for Re<2100

f=16/Re

The results for these

empirical equations are

presented in the Moody chart.

This allows friction factor(f) to

be determined once Re and k/d are known.

2

42u

D

fLP

Pressure loss in a pipe (generated by equating

force due to shear-stress to force due to

pressure-drop) given by Darcy-Weisbach

equation:

∆P=Pressure drop

f=friction factor

L=length of pipe

ρ=density

u=mean velocity

D=diameter of pipe

Internal flows – Moody chart

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Turbulent flow

Blasius, f=0.079/Re0.25

Internal flows – Pipe and Duct pressure losses and flows

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(three methods of

investigating or

sizing pipes)

References

• Fluid Mechanics – Douglas, Gasiorek, Swaffield (Published Longman Scientific & Technical)

• Introduction to Fluid Mechanics – Shaughnessey, Katz, Schaffer (Published Oxford University Press)

• Race Car Aerodynamics, Designing for Speed – Joseph Katz (Published Robert Bently)

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