ME 3560 Fluid Mechanics - Homepages at...

ME3560 – Fluid Mechanics

Chapter IX. Flow over Immersed Bodies

Spring 2018

1

ME 3560 Fluid Mechanics




Spring 2018

2

9.1 General External Flow Characteristics

•A body immersed in a moving fluid experiences a resultant force due tothe interaction between the body and the fluid surrounding it.

•In some instances the fluid far from the body is stationary and the bodymoves through the fluid with velocity U.

•Or the body is stationary and the fluid flows past the body withvelocity U.

•In both cases, the coordinate system can be fixed to the body and treatthe situation as fluid flowing past a stationary body with velocity U,the upstream velocity.



Spring 2018

3

•The structure of an external flow and the description and analysis of theflow depend on the nature of the body in the flow.

•Two-dimensional objects infinitely long and of constant cross-sectional size and shape.

•Axisymmetric bodies formed by rotating their cross-sectional shapeabout the axis of symmetry.

•Three-dimensional bodies that may or may not possess a line or plane ofsymmetry.



Spring 2018

4

9.1.1 Lift and Drag•When any body moves through a fluid, two types of forces occur as aresult of this motion:•Wall shear stress (τw) forces, due to viscous effects•Normal stress forces, due to the pressure, p.

•The resultant force in the direction ofthe upstream velocity is termedthe drag, D, and the resultant forcenormal to the upstream velocity istermed the lift, L•For some three-dimensional bodiesthere may also be a side force that isperpendicular to the plane containing Dand L.



Spring 2018

5

•The resultant of the shear stress and pressure distributions can beobtained by integrating the effect of these two quantities on the bodysurface:

cos)(sin)(sin)(cos)(

dApdAdFdApdAdF

wy

wx

dAdApdFL

dAdApdFD

wy

wx

cossin

sincos

•To carry out the integrations and determine the lift and drag, the bodyshape, the distribution of τw and p along the surface need to be known.• These distributions are often extremely difficult to obtain.•The pressure distribution can be obtained experimentally by use of aseries of static pressure taps along the body surface.• It is usually quite difficult to measure the wall shear stress distribution



Spring 2018

6

The equations to determine drag and lift,

dAdApLdAdApD ww cossinsincos

are valid for any body, however, it is necessary to know the appropriate shear stress and pressure distributions on the body surface. This is not easy, such information is available only for certain simple situations.

Drag (viscous)Friction

sin

DragPressure

cos dAdApD w

NegligibleUsually

cossin dAdApL w



Spring 2018

7

Alternatively, dimensionless lift and drag coefficients are defined and their approximate values are determined by means of either a simplified analysis, some numerical technique, or an appropriate experiment. The lift coefficient, CL , and drag coefficient, CD, are defined as

AULC

AUDC LD 2

212

21

A is a characteristic area of the object. U is the upstream velocity.



Spring 2018

8

Pressure-Sensitive Paint (PSP)

PSP is now gaining acceptance as an alternative to the static surface pressure ports. The PSP material is typically a luminescent compound that is sensitive to the pressure on it and can be excited by an appropriate light which is captured by special video imaging equipment. Thus, it provides a quantitative measure of the surface pressure. PSP is a global measurement technique, measuring pressure over the entire surface, as opposed to discrete points. PSP also has the advantage of being nonintrusive to the flow field.



Spring 2018

9

Particular Cases:•θ = 0.

•Friction Drag = 0. Drag is due topressure only (form drag).

•θ = π/2. Flat plate parallel to flow

•Pressure Drag = 0. Only viscousdrag is present.



Spring 2018

10

9.1.2 Characteristics of Flow Past an Object

lU

Re



Spring 2018

11

DU

Re



Spring 2018

12

9.2 Boundary Layer Characteristics•Consider the case in which the boundary layer is formed on aninfinitely long flat plate along which flows a viscous, incompressiblefluid.

xU

Re



Spring 2018

13

•Some distance downstream from the leading edge, the boundary layer flow becomes turbulent. •Turbulent flow is characterized by the occurrence of irregular mixing of fluid particles that range in size from the smallest fluid particles up to those comparable in size with the object of interest.•For laminar flow, mixing occurs only on the molecular scale.



Spring 2018

14

•The transition from a laminar boundary layer to a turbulent boundary layer occurs at a critical value of the Reynolds number,

Rexcr= 2 × 105 to 3 × 106

•Depending on the surface roughness and the amount of turbulence in the upstream flow. •The location along the plate where the flow becomes turbulent, xcr, moves towards the leading edge as the free-stream velocity increases.



Spring 2018

15

•The boundary layer thickness, δ, is the distance from the plate at which the fluid velocity is 0.99U: Uuy 99.0where

• Fig. b shows two velocity profiles—one if there were noviscosity (a uniform profile) and one considering the viscosityand zero slip at the wall (the boundary layer profile).



Spring 2018

16

Laminar Boundary Layer:

xUw

2/3332.0

Rexcr = 5×105

xUx

orU

xx

x

Re;Re55

221 U

C wf

xfC

Re664.0

LDf

DDf C

AUFC

Re328.1

221

If Rex < 5×105



Spring 2018

17

9.2.4Transition from Turbulent to Laminar Flow•The analytical results shown in the previous section are restricted to laminar boundary layer flows along a flat plate with zero pressure gradient.

•They agree quite well with experimental results up to the point where the boundary layer flow becomes turbulent.

•Any BL will become turbulent for any free-stream velocity and any fluid provided the plate is long enough.



Spring 2018

18

9.2.4Transition from Turbulent to Laminar Flow•Re at the transition location is a function of various parameters:- Roughness of the surface.- The curvature of the surface (for example, a flat plate or a sphere).- The disturbances in the flow outside the boundary layer.

•On a flat plate with a sharp leading edge in a typical airstream, the transition takes place at Rexcr = 2 × 105 to 3 × 106. •For our Calculations we will use Rexcr = 5×105



Spring 2018

19

•The transition from laminar to turbulent flow involves the instability of the flow field.

•Small disturbances imposed on the boundary layer flow: from a vibration of the plate, a roughness of the surface will either grow (instability) or decay (stability), depending on where the disturbance is introduced into the flow.

•If these disturbances occur at a location with Rex < Rexcr they will die out, and the boundary layer will return to laminar flow at that location.

•Disturbances imposed at a location with Rex > Rexcr will grow and transform the boundary layer flow downstream of this location into turbulence.



Spring 2018

20

•The transition from laminar toturbulent flow involves anoticeable change in the shape ofthe boundary layer velocityprofile.

•The turbulent profiles are flatter,have a larger velocity gradient atthe wall, and produce a largerboundary layer thickness than dothe laminar profiles.



Spring 2018

21

9.2.5 Turbulent Boundary Layer Flow•The structure of turbulentboundary layer flow is verycomplex, random, and irregular.

•The figure shows a laser-inducedfluorescence visualization of aturbulent boundary layer on a flatplate.

•There are no “exact” solutionsfor turbulent boundary layer flow.•Considerable effort has beenmade to obtain numeric solutionsfor turbulent flow by usingapproximate shear stressrelationships.



Spring 2018

22

Turbulent Boundary Layer:

xUx

orxU xx

Re;Re370.0370.0

5/15/45/1

If Rex > 5×105

Rexcr = 5×105

5/12 Re0288.0 xw U

5/1221 Re

072.0

L

DDf AU

FC

5/1Re0576.0 xfC

221 U

C wf



Spring 2018

23

In Summary:Laminar Boundary Layer:

xUw

2/3332.0

Rexcr = 5×105

xUx

orU

xx

x

Re;Re55

221 U

C wf

xfC

Re664.0

LDf

DDf C

AUFC

Re328.1

221

If Rex < 5×105



Spring 2018

24

Turbulent Boundary Layer:

xUx

orxU xx

Re;Re370.0370.0

5/15/45/1

If Rex > 5×105

Rexcr = 5×105

5/12 Re0288.0 xw U

5/1221 Re

072.0

L

DDf AU

FC

5/1Re0576.0 xfC221 U

C wf



Spring 2018

25

9.3 Drag

AUDCD 2

21



Spring 2018

26

9.4 Lift

AULCL 2

21

Typically, the lift is given in terms of the lift coefficient

•Most common lift-generating devices (i.e., airfoils, fans, spoilers on cars, etc.) operate in the large Re range in which the flow has a boundary layer character, with viscous effects confined to the boundary layers and wake regions.• For such cases the wall shear stress, τw, contributes little to the lift.•Most of the lift comes from the surface pressure distribution. •The distribution, for the most part, is consistent with simple Bernoulli equation analysis. •Locations with high-speed flow have low pressure, while locations with low-speed flow have high pressure.



Spring 2018

27

•A typical device designed to produce lift does so by generating apressure distribution that is different on the top and bottom surfaces.

•For large Reynolds number flows these pressure distributions areusually directly proportional to the dynamic pressure, ρU2/2, withviscous effects being of secondary importance.

•Two airfoils used to produce lift are shown. •The symmetrical one cannot produce lift unless the angle of attack, α, is nonzero.

•Because of the asymmetry of the nonsymmetric airfoil, the pressure distributions on the upper and lower surfaces are different, and a lift is produced even with α = 0.



Spring 2018

28

•Since most airfoils are thin, it is customary to use the planformarea, A = bc, in the definition of the lift coefficient.

• b is the airfoil span and c is the chord length—the length from theleading edge to the trailing edge.

•Typical lift coefficients so defined are on the order of unity. That is, thelift force is on the order of the dynamic pressure times the planform areaof the wing, L ≈ (ρU2/2)A.

•The wing loading, defined as the average lift per unit area of thewing, L/A, therefore, increases with speed.



Spring 2018

29

•Typical lift and drag coefficient data as a function of angle of attack, α,and aspect ratio, AR, are shown.•AR is the ratio of the square of the wing span to the planform area,AR = b2/A.•If the chord length, c, is constant along the length of the wing (arectangular planform wing), AR= b/c



Spring 2018

30

•In general, CL increases and CD decreases with an increase in AR.

• Long wings are more efficient because their wing tip losses arerelatively minor than for short wings.

•The increase in drag due to the finite length (AR < ∞) of the wing isoften termed induced drag.

•It is due to the interaction of the complex swirling flow structure nearthe wing tips and the free stream.



Spring 2018

31

•Although viscous effects and the wall shear stress contribute little to thedirect generation of lift, they play an extremely important role in thedesign and use of lifting devices.

•This is because of the viscosity-induced boundary layer separation that can occur on nonstreamlined bodies such as airfoils that have too large an angle of attack



Spring 2018

32

•As is indicated in the figure, up to a certain point, CL increases rathersteadily with the angle of attack.

•If α is too large, the boundary layer on the upper surface separates, theflow over the wing develops a wide, turbulent wake region, the liftdecreases, and the drag increases.

•This condition, as indicated by thefigures in the margin, is termedstall.

ME 3560 Fluid Mechanics - Homepages at...

Documents

Transcript of ME 3560 Fluid Mechanics - Homepages at...