9 Ch 9 - External incompressible viscous flow v1 PDF

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4/9/2018

External incompressible viscous flow Now we’ll have a look at aspects of external incompressible viscous flows, which are characterized by unbounded growth of a boundary layer. - The boundary layer (12) - Drag (9) - Lift (6)

Mark McQuilling

ESCI 3200 Fluid Dynamics

v 1.1

Saint Louis University

Slide 1

The boundary layer (1) Let’s consider flow over an airfoil as shown below:

• flow divides at the stagnation point, while boundary layers grow on both sides - geometry determines the pressure gradients, which in turn influence boundary layer growth and laminar vs turbulent characteristics - depending on the above, viscous wakes can develop behind the airfoil and greatly influence drag forces (wakes = low pressure behind object = more drag) - possible separation bubbles can also influence drag and lift forces • since curvature in geometry determines the pressure gradients which have a very significant influence on flow physics, we will at first limit our discussion of boundary layers to flat plates with zero streamwise pressure gradients Mark McQuilling

ESCI 3200 Fluid Dynamics

v 1.1

Saint Louis University

Slide 2

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4/9/2018

The boundary layer (2) The boundary layer concept was the missing piece of information that allowed theoretical fluid mechanics to match experiment. In 1904, Ludwig Prandtl showed how you can analyze fluid flow by dividing the fluid into two regions: one close to the surface and one far enough away from the surface. The region close to the surface is called the boundary layer, where effects of viscosity are important (i.e. requires the Navier-Stokes equations). Outside the boundary layer, viscous effects are negligible and can be treated as inviscid (using Euler’s equation). • both viscous and momentum forces are important, so the Reynolds number is often used to characterize boundary layers - distance from the leading edge is most often used as the length scale • a simple flat plate flow has constant outer velocity U, and therefore constant pressure gradient

• a typical flat plate boundary layer under zero pressure gradient can transition to turbulence after Re x  500,000

Mark McQuilling

ESCI 3200 Fluid Dynamics

v 1.1

Saint Louis University

Slide 3

The boundary layer (3) In 1921, Theodore von Kármán (1881-1963) applied the integral forms of our governing equations to a flat plate boundary layer flow and developed relations that are some of the most widely used descriptors of boundary layer “physics”. Here’s how that goes:

streamline outside shear area

U

y Y

U

yH

boundary layer height,  u 0 .99U c.v.

u  u( y )

y x

no-slip wall

drag force, D

• Now we apply the integral conservation of mass:





Y

H

0

0

  V  dA  0   udy   Udy cs Mark McQuilling

ESCI 3200 Fluid Dynamics

v 1.1

Saint Louis University

Slide 4

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4/9/2018

The boundary layer (4) Assuming incompressible flow gives: Y

Y

UH   udy   (U  u U )dy 0

0

Y

 UY   ( u  U ) dy 0

* * • Now define Y  H      Y  H to get: Y

U( Y  H )  U *   (U  u) dy 0

Y 

*   0

1 u    dy  U

• This is called the displacement thickness

http://aerojockey.com/papers/bl/img11.gif

* • Streamlines outside  u0 .99U will deflect an amount  ( x) , which acts similar to a free-slip wall - represents distance a surface would need to move vertically parallel to itself to produce an inviscid mass flow rate that is the same as the real case * -  ( x ) represents how much the effective flow area changes with downstream distance

• This definition is good for laminar and turbulent flows, but not for separated flows! Mark McQuilling

ESCI 3200 Fluid Dynamics

v 1.1

Saint Louis University

Slide 5

The boundary layer (5) Let’s apply the integral conservation of momentum on our control volume:   Y H  Fx   D   u (V  dA )   u (udy )  U (U )dy 0

0

cs Y

Drag  D   U 2 H    u 2dy 0

• Now let’s Yassume incompressible flow and use UH   udy from earlier to get: 0

Y

D    u(U  u )dy 0

D

U 2

  

Y 

0

u u 1  dy U U

• This is called the momentum thickness http://nptel.ac.in/courses/1 01103004 /module5/lec9/images/41.pn g

• The momentum thickness applies to laminar and turbulent incompressible boundary layers, 2 but if the geometry is not a flat plate, then   D / U . • represents the distance a surface would have to be moved parallel to itself in an inviscid fluid to give the same total momentum as that occurring in a real fluid • not applied to separated flow Mark McQuilling

ESCI 3200 Fluid Dynamics

v 1.1

Saint Louis University

Slide 6

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4/9/2018

The boundary layer (6) The shape factor is defined as: H

• • • • •

* 

this is not the same height (H) used in the derivations of  * and  used to describe “character” of flow, and is always > 1 H increases with stronger adverse pressure gradients H = 2.59 for Blasius flow (laminar flat plate with no pressure gradient) H = 1.2-1.4 for a fully turbulent boundary layer

• The wall shear stress  w (x ) is related to the flat plate drag by: • this is a per unit depth drag; otherwise, the integration would cover a two-dimensional space

L

D   w ( x ) dx 0

• Or in non-dimensional form: • friction coefficient:

C f ( x) 

• drag coefficient:

CD 

Mark McQuilling

d  w (x ) 2 0.5 U 2 dx • a 3D drag coefficient would include an area instead of a length in the denominator

D 0.5U 2L

ESCI 3200 Fluid Dynamics

v 1.1

Saint Louis University

Slide 7

The boundary layer (7) The previous relations are exact with the correct u(y); we can even get pretty close with a reasonable guess, as long as u(y) satisfies the boundary layer boundary conditions: 1. no-slip wall: u (0)  0  2. smooth merge with freestream: u(  )  U , u  0 y y  • A second-order polynomial can satisfy these:

  y   y  2 u  U 2         δ   δ  

2δ 5 δ , H δ*  ,   3 15 2 15 μdx 2 μU / δ d d  2  d   2 2  Cf   2 U 0.5U dx dx  15 

• Substitution into the definitions for δ* , gives:

 w 

u 2 μU  δ  y y 0

2

30 μx U

 x



5.5 Rex

• This is Kármán’s estimate (much easier than N-S)

• Prandtl also conducted an order of magnitude analysis and found: v  U , P  P ( x ) - means you can measure static pressure at surface and assume it’s constant normal to the surface (at least through the boundary layer) - allows Euler’s equation to determine freestream (and surface) pressure distribution Mark McQuilling

ESCI 3200 Fluid Dynamics

v 1.1

Saint Louis University

Slide 8

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The boundary layer (8) Table 9.2 shows various results of boundary layer parameters based on various estimates of boundary layer velocity profiles:

• these results are for zero pressure gradient!

• the exact solution noted in Table 9.2 was found for laminar flow by Blasius, where the others are called “approximate solutions” because they require an estimated velocity profile shape based on polynomials • Fig. 9.7 at right shows how turbulent velocity profiles are much fuller and have more nearwall momentum Mark McQuilling

ESCI 3200 Fluid Dynamics

v 1.1

Saint Louis University

Slide 9

The boundary layer (9) • most of the time we want to know the boundary layer height and the drag force on the surface * -  and  are often used to correlate various phenomena (transition, separation, etc.) over a wide range of conditions ( laminar vs turbulent, w/ and w/o P / x , etc.) -  * also represents effective area constriction due to growth of boundary layer

• Fig. 9.6 illustrates how pressure gradients can greatly modify the zeropressure gradient “flat plate” behavior

• favorable pressure gradients P / x  0 lead to thinner boundary layers and influence the flow towards a laminar state due to the acceleration • adverse pressure gradients P / x  0 lead to thicker boundary layers and influence the flow towards separation or turbulence (depending on local flow momentum) Mark McQuilling

ESCI 3200 Fluid Dynamics

v 1.1

Saint Louis University

Slide 10

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4/9/2018

The boundary layer (10) • combining integral conservations of mass and momentum on any dU w d (U 2 )  *U  boundary layer flow leads to the momentum integral equation:  dx dx - for known u / U and  w this equation can be integrated to * find  , and then we can find  ,  _ - expanding the 1st term on the RHS gives: dU d  w C f d  dU w    (H  2) U 2  (  *  2 ) U dx U dx dx dx U 2 2  • for turbulent flow following a 1/7th power law profile, the momentum integral equation produces the following for boundary layer height and skin friction coefficient: 1/ 7 w   0. 382 0. 0594 u y , Cf   Cf     0 .5U x Re1x/ 5 Re1x/ 5 U    - experiments confirm the accuracy of the above for 5  105  Re x  1  107

2

  

• applying the momentum integral equation to laminar and turbulent velocity profiles shows turbulent boundary layers grow more quickly than laminar ones (due to the increased mixing!):  turb ~ x 4 / 5  lam ~ x 1/ 2 - although these are for P / x  0 , the same trends hold true for all flows Mark McQuilling

ESCI 3200 Fluid Dynamics

v 1.1

Saint Louis University

Slide 11

The boundary layer (11) Example problem 30 (Example 9.2): Consider two-dimensional laminar boundary layer flow along a flat plate. Assuming the velocity profile is sinusoidal, find expressions for (a) the rate of * growth of  ( x) , (b) the displacement thickness  (x ), and (c) the total friction force on the plate of length L and width b. u  y   sin   U 2  

Mark McQuilling

ESCI 3200 Fluid Dynamics

v 1.1

Saint Louis University

Slide 12

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4/9/2018

The boundary layer (12) Example problem 30 (continued)

Mark McQuilling

ESCI 3200 Fluid Dynamics

v 1.1

Saint Louis University

Slide 13

Drag (1) Any time there is relative motion between a solid body and a viscous fluid, the body will experience a net force which depends on the geometry and fluid flow properties and conditions. We can generalize the resultant forces into drag and lift, where drag is the resultant force in the direction of motion (or that opposing motion), and lift is the resultant force in the direction perpendicular to motion. • both drag and lift are determined by integration of normal and shear stresses on the surfaces Re=0.16

Re=26

Re=10,000

van Dyke, Album of Fluid Motion

• drag and lift are at least a function of Reynolds number, and it also correlates with laminar vs turbulent conditions (which can change the normal and shear stresses) - three images above are for circular cylinder flows, showing wake structures = f( Re) - larger wakes = lower pressures = ↑ drag - streamlined vs bluff body geometry aspects also influence wake structure Mark McQuilling

ESCI 3200 Fluid Dynamics

v 1.1

Saint Louis University

Slide 14

7

4/9/2018

Drag (2) When we considered Buckingham PI analysis for the drag of a sphere in cross flow, the result contained a non-dimensional group including the drag force which we’ll call the drag coefficient: CD 

FD 1 V 2 A 2

• we added a ½ to use the dynamic pressure in the denominator

• for the smooth sphere we had C D  f (Re) , but it could also include compressibility or freesurface effects: C D  f (Re, Fr, M ) • for flow over a parallel flat plate, we have pure friction drag:



FD 



w surface

dA

• for laminar flow we already had C f  CD  Mark McQuilling

CD 

w 0 .5 U2

1 0. 664 Rex1/ 2 dA A A



w plate surface

dA

1 V 2A 2

0 .664 so that: Rex

CD 

ESCI 3200 Fluid Dynamics

v 1.1

1.33 Re L Saint Louis University

Slide 15

Drag (3)  u  y1/ 7  0. 0594 • a turbulent boundary layer      with C f  leads to: Re 1x/ 5 U   

CD 

0. 0742 Re1L/ 5

(parallel flat plate only, 5 10 5  Re L 1 10 7)

• Schlichting’s Boundary Layer Theory gives this experimental correlation: CD 

0. 455 (log Re L )2 .58

(parallel flat plate only, ReL  1 10 9)

• if the flat plate flow starts laminar and transitions to turbulence, use these: CD 

CD 

Mark McQuilling

0. 0742 1740  ReL Re1L/ 5

0. 455 1610  2 .58 Re L (log Re L ) ESCI 3200 Fluid Dynamics

(parallel flat plate only, 5 10 5  Re L 1 10 7)

(parallel flat plate only, Re L 1 10 9)

v 1.1

Saint Louis University

Slide 16

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4/9/2018

Drag (4) • for flow into a flat plate, we have pure pressure drag with: FD   pdA surface - shear stress makes no contribution in streamwise direction - bluff body aerodynamics - sharp points make drag less dependent on Reynolds number (after Re ~ 1000 )

• drag coefficient usually based on frontal projected area - for airfoils and wings, the planform area is used

• Fig. 9.10 shows parallel flat plate drag coefficient vs aspect ratio (width to height)

Mark McQuilling

ESCI 3200 Fluid Dynamics

v 1.1

Saint Louis University

Slide 17

Drag (5) In most cases, both friction and pressure forces cause net drag; Table 9.3 shows drag coefficients for a variety of selected objects; keep in mind there are lots of references that provide such info

Mark McQuilling

ESCI 3200 Fluid Dynamics

v 1.1

Saint Louis University

Slide 18

9

4/9/2018

Drag (6) Here’s the drag coefficient for a smooth circular cylinder vs Reynolds number:

• with increasing Reynolds number: - laminar boundary layer gives way to steady separation bubbles in the wake - then wake bubbles oscillate (causing oscillating forces), wake transitions to turbulence - then the flow transitions to turbulence, and the separation point moves downstream Mark McQuilling

ESCI 3200 Fluid Dynamics

v 1.1

Saint Louis University

Slide 19

Drag (7) Here’s the drag coefficient for a smooth sphere vs Reynolds number:

• drag coefficient for a turbulent boundary layer is ~ 1/5 of that in a laminar boundary layer - lots of research and development money is spent on golf ball dimple design • surface roughness – like dimples – can significantly alter the flow physics, as shown in the movie at right Mark McQuilling

ESCI 3200 Fluid Dynamics

v 1.1

Saint Louis University

Slide 20

10

4/9/2018

Drag (8) Several approaches can allow a reduction in drag, such as when two or more objects move in tandem so their flowfields interact; bicycle riders and race car drivers employ this technique all the time – it’s called drafting:

https://thescienceclassroom.wikispaces.com/file/view/drafting.pn g/225476 104/drafting .png

• streamlining also reduces net drag by trading pressure drag for friction drag - the adverse pressure gradient towards the trailing end of a body can be spread out over a longer distance Mark McQuilling

ESCI 3200 Fluid Dynamics

http://blogsdir.cms.rrcdn.com/1 0/files/2013/11/ATDynamics-TrailerTail.jpg v 1.1

Saint Louis University

Slide 21

Drag (9) Example problem 31 (Example 9.5): A cylindrical chimney 1m in diameter and 25m tall is exposed to a uniform 50km/hr wind at standard atmospheric conditions. End effects and gusts can be neglected. Estimate the bending moment at the base of the chimney due to the wind.

Mark McQuilling

ESCI 3200 Fluid Dynamics

v 1.1

Saint Louis University

Slide 22

11

4/9/2018

Lift (1) Some objects are designed to generate lift, and we define a non-dimensional lift coefficient:

CL 

FL 1  V2 A 2

• lift coefficients are functions of Reynolds number and angle of attack,  • although most objects in a flow generate (positive or negative) lift, airfoils are devices used specifically to generate positive lift - the chord of the airfoil is a straight line connecting the leading and trailing edges - most wing shapes are defined with a mean line and a thickness distribution - cambered airfoils have incongruent chord and mean lines - lift is generated due to the pressure difference on the top and bottom surfaces - three movies below show flow past an airfoil at  = 0°, 10°, and 20°

Mark McQuilling

ESCI 3200 Fluid Dynamics

v 1.1

Saint Louis University

Slide 23

Lift (2) Fig. 9.17 shows lift and drag coefficients vs angle of attack for two airfoil sections with 15% thickness ratio (of percent chord) at Re=9,000,000 - National Advisory Committee for Aeronautics (NACA) naming convention: conventional - 2

30

15

thickness 15 % chord

max camber location

 1   30  15 %chord  2

design lift coefficient  3   0.2  0.3   2

laminar flow - 66

2

2

15

thickness 15 % chord

design lift coefficient (0.2) max lift coefficient, dp/dx...