flow past immersed bodies

ENSC 283

INTRODUCTION TO FLUID MECHANICS

Chapter 7 - Flow Past Immersed Bodies

Boundary Layer Theory

External Flows of Incompressible Viscous Fluids

Peyman Taheri

ENSC 283 (Spring 2013) 2

Introduction

In Chapter 6, we studied flows completely bounded by solid surfaces, so-called “internal flows”. In this chapter, we use, and extend, the concepts we learned for internal flows, to study external flows, in which flow is partially bounded by solid surfaces.

Similar to the internal flows, there are three techniques to study external flows: i) numerical solutions with computers, ii) experimentation, and iii) analytical solutions with boundary layer theory. In this course we must focus on the third tool, boundary layer theory, which was first formulated by Ludwig Prandtl in 1904.

Similar to Chapter 6, in this section we will only consider viscous incompressible flows; hence we will study the flow of liquids and gases that have negligible heat transfer, for which the Mach number is small, Ma 0.3< .

Reynolds Number and Geometry Effects

Boundary layer analysis can be used to compute viscous effects near solid walls, and to “patch” them onto the outer inviscid flow. This patching technique is more successful as the Reynolds number becomes larger.

In Fig. 1, a uniform stream with velocity U moves parallel to a sharp flat plate of length L . If the Reynolds number (based on the length of the plate) Re /L ULρ μ= is small, as shown in Fig. 1, the viscous region is very broad and extends far ahead, and to the sides of the plate. The plate retards the oncoming stream greatly, and small changes in stream flow properties cause large changes in the pressure distribution along the plate (observed in experiments). Thus, although in principle it should be possible to patch the viscous region (inside the boundary layer) and inviscid region (outside the boundary layer) in a mathematical analysis, it turns to be a complicated analysis, due to strong nonlinearity1.

Note: Thick-shear-layer flows (thick boundary layers) exist for Re 1000L < , for which numerical or experimental modeling are appropriate.

In contrast, a high Reynolds number flow, as shown in Fig. 2, is more suitable to boundary layer patching. The viscous layer, either laminar or turbulent, is very thin and its coupling to the inviscid layer is almost linear.

We define the boundary layer thickness δ as the locus of points where the velocity in the viscous layer parallel to the plate u reaches 99 percent of the inviscid free stream velocity U . In the proceeding sections we will see that for laminar and turbulent flows over a “flat-plate” the thickness of boundary layer at point x can be calculated from the following formulas, obtained from the exact solution of momentum equation;

1 When two processes are coupled nonlinearly, it means that even a very small change in one process can cause a drastic (and sometimes unpredictable) change in the other. In contrast, linear coupling of two processes means that the rate of their changes is a constant.

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3 61/2

61/7

5 laminar 10 Re 10Re0.16 turbulent 10 ReRe

xx

xx

xδ

⎧ < <⎪⎪= ⎨⎪ <⎪⎩

(1)

Figure 1: Flow past a sharp flat plate at low Reynolds number.

Figure 2: Flow past a sharp flat plate at high Reynolds number.

where the “local Reynolds number” is,

RexUxρμ

=

Note: As mentioned above, for Re 1000x < boundary layer theory is not applicable.

For slender bodies, such as plates and airfoils parallel to the oncoming stream, and for sufficiently large Reynolds numbers, we conclude that the interaction of viscous layer and inviscid regions is linear and most of the times negligible. However, for blunt bodies, even at very high Reynolds number flows, there is a discrepancy in the viscous-inviscid patching concept. Figure 3 shows two sketches of flow past a cylinder (or sphere). In sketch (a) the idealized case is shown in which there is a thin film of boundary

ENSC 283

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Peyman Taheri


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Momentum Integral Estimates for Flat Boundary Layer

In this section a simple control volume analysis is introduced to find an expression for the drag force on the surface.

As depicted in Fig. 4, a free stream with uniform velocity U=V i is parallel to a flat plate of length L and width b . A boundary layer (either laminar or turbulent) with thickness δ at x L= is formed. The measurements show that pressure is almost constant inside the boundary layer. The viscous stress along the plate yields a finite drag force on the plate.

Figure 4: Growth of a boundary layer over a flat plate.

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We shall consider a streamline which meets the boundary layer at x L= . A control volume can be constructed using this streamline. The control volume includes the boundary layer and a portion of the free stream. The boundaries of the control volume are labels from 1 to 4 in Fig. 4.

The x-momentum equation for incompressible flows in integral form is,

ddxFt

=∑outlets inletsCV

d wherem m m VAρ ρV V V⎛ ⎞

+ − =⎜ ⎟⎝ ⎠

∑ ∑∫ V (2)

where the first term vanished due to steady-state condition.

The only force in x-direction is the “drag force” D, due to shear on the wall, which is acting on the wall in positive x-direction, but the wall inserts the same force on the fluid in opposite (negative) direction. Since pressure is constant, its net force is zero. Also body forces due to gravity are not acting in x-direction.

The inlet for the control volume is boundary 1, and the outlet is boundary 3. Indeed, no flow passes through the streamline (boundary 2) and wall (boundary 4). Accordingly, Eq. (2) reads,

[ ] ( )

boundary 3 boundary 1

boundary 3 boundary 1

( , ) ( , ) d d

x xD V m V m

u L y u L y b y U Ub yρ ρ

− = −

= −

∫ ∫

∫ ∫ (3)

Note that at boundary 3 the velocity is ( , )u L y and at boundary 1 the inlet velocity is U . The integrals can be calculated as,

2 2 2 2

0 0 00

d d dh

y y yx L x x L

D b u y U b y b u y U bhδ δ

ρ ρ ρ ρ= = == = =

− = − = −∫ ∫ ∫ (4)

The value of h is unknown, and we must use the mass conservation equation to find it, i.e.,

outlets inlets

where i i i im m m V Aρ= =∑ ∑ (5)

or,

ρ u b0

dy x L

yδ

ρ= =

=∫ U b0 00

d dh

y yx x L

y u y Uhδ

= == =

→ =∫ ∫ (6)

Substitution of Uh from Eq. (6) into the last term of Eq. (4) gives,

2

0 0

d dy yx L x L

D Ub u y b u yδ δ

ρ ρ= == =

= −∫ ∫ (7)

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or,

( )0

dy x L

D b u U u yδ

ρ= =

⎡ ⎤⎢ ⎥= −⎢ ⎥⎢ ⎥⎣ ⎦∫ (8)

Note: The above equation for “drag force” was first derived by von Karman in 1921, who wrote it in the convenient form of the “momentum thickness” θ ,

( )

2

0

( ) with 1 dx u uD x bU y

U U

δ

ρ θ θ⎛ ⎞⎟⎜= = − ⎟⎜ ⎟⎜⎝ ⎠∫ (9)

Note: As an assignment you will prove that shear stress on the wall wτ can be approximated from von Karman’s relation (9) as,

( ) 2w

dd

x Uxθτ ρ=

which is in general valid for both laminar and turbulent flows. For laminar flow, it can be shown that von Karman’s relation gives,

1/2lam

5.5Rexx

δ= (10)

which predicts the thickness of boundary layer 10% higher than the “exact solution” in Eq. (1).

Exact Solution for Momentum Boundary Layer

Exact solution of momentum equations for boundary layer over a flat surface is discussed in Sections 7.3 and 7.4 of the textbook. Essentially, this exact solution, known as “Blasius solution”, can be obtained by simplifying the x and y momentum equations, and solving them by defining a “similarity variable”.

Note: The relations given in Eq. (1) are obtained from Blasius exact solution.

For this course the details of Blasius solution is not covered and we must restrict ourselves to the final results, which are useful to solve the problems, as listed below.

Laminar flow

D1/2 1/2 2 1/2

5 0.664 2 ( ) 1.328 2 ( )Re Re Ref f

x x L

D Lc C c Lx U bLδ

ρ= = = = = (11)

where δ is the thickness of boundary layer, fc is the “skin friction factor”, Rex is the local Reynolds

number, ReL is the Reynolds number at x L= , DC is the “drag coefficient”, ( )D L is the drag force on the plate of length L. Width of the plate is b; free stream density and velocity are ρ and U , respectively.

ENSC 283

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Peyman Taheri


Blunt-body External Flows

In flows past blunt bodies, due to separation of boundary layer (See. Fig. 3b), the theory is unable to give satisfactory analysis. Accordingly, such flows which are common in aerodynamics and flight applications are mostly investigated experimentally and numerically.

Note: In parallel flows past a slender body (e.g., a flat-plate as shown in Fig. 4) the only force exerted to the body is the “drag” force due to shear stress at the surface. However, in flows pas a blunt body, another force may result which is called “lift” force.

The first human flight was by Montgolfier brothers in 1783 with a hot-air balloon over Paris (about 6 miles). Flight with a balloon, which is known as lighter-than-air craft, is the results of buoyant force, and thus, the balloon flight technology has contributed nothing to powered-flight technology for heavier-than-air craft. The powered-flight technology is based on the fundamental idea of moving inclined surfaces in the flight direction to generate lift force.

Forces of Flight (Aerodynamic Forces)

Aerodynamic forces are “lift” and “drag”, which result from movement of a body in a fluid, see Fig. 6. The sources of these forces are shear stresses (viscous effects) and normal stresses (pressure effects).

Figure 6: Aerodynamic forces (lift and drag) and other forces acting on a flying eagle.

For an airfoil, distribution of pressure and shear stress on its surface A are schematically shown in Fig. 7. The negative pressures in the pressure distribution sketch means negative with respect to atmospheric pressure (negative gage pressure). The total force on the airfoil is the summation of pressure and viscous forces, i.e.,

total wd dA A

F p A Aτ= +∫ ∫ (14)

This total force can be divided into two components: a) lift force which is normal to the free stream velocity, and b) drag force which is parallel to the free stream velocity.

Note: The calculation of lift and drag forces are possible by computer simulations (Computational Fluid Dynamics) and/or experimentation.

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Figure 7: Schematic distribution of normal stress (pressure) and shear stress (viscosity effects) are shown for an airfoil.

Lift and Drag Forces and the Coefficients

Let’s consider a small elemental area on an airfoil as shown in Fig. 8. Components of the fluid forces in x and y directions are,

( ) ( ) ( ) ( )w wd d cos d sin d d sin d cosx yF p A A F p A Aθ τ θ θ τ θ= + =− + (15)

The velocity is parallel to the x axis, then, according to the definition, drag force is in x direction and lift force is in y direction,

( ) ( )w wd d cos d sin d sin d cos dx yA A A A A A

D F p A A L F p A Aθ τ θ θ τ θ= = + = =− +∫ ∫ ∫ ∫ ∫ ∫ (16)

To compute the above integrals, we need a complete set of geometrical information, i.e., the value of θ at every point of the surface. Even more, pressure and shear at every point must be known.

Figure 8: Normal stress (pressure) and shear stress (viscosity effects) on an elemental surface area (dA) of an airfoil.

Lift and drag coefficients are dimensionless quantities defined as,

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2 2

drag force lift force1 1dynamic pressure Area dynamic pressure Area2 2

D LD LC CU A U Aρ ρ

= = = =× ×

(17)

where 2 /2Uρ is the dynamic pressure. The coefficients DC and LC strongly depend on the geometry of the object, and hence are usually determined by experiment/simulation. For example, lift coefficient of airfoils can be increased significantly with small change in their orientation (angle of attack) and geometry, e.g., adding flaps and slats (see Fig. 9).

Figure 9: Effects of high-lift devices, i.e., flaps and slats on lift coefficient of airfoils.

For some two-dimensional, and three-dimensional bodies the drag coefficient are listed in Table 1 and 2.

Wake and Separation

In flows past blunt bodies, the point at which boundary layer breaks off is called the “separation point”. In Fig. 10, effects of separated flow and the subsequent failure of boundary layer theory are illustrated. Note that separation and wake is not restricted to turbulent flows only; indeed separation and wake can exist in laminar flow as well, see Fig. 11. Nonetheless, the size and structure of wakes are different in laminar and turbulent flows.

Figure 10: Flow around a blunt body at different Reynolds numbers. Left) at very small Reynolds numbers, a creeping flow is observed and flow is completely laminar around the object. Center) for flow around a cylinder, at moderate Reynolds numbers, around Re 20= , separation of laminar boundary layer starts and separation bubbles are formed. Right) at high Reynolds number the turbulent flow behind a tennis ball is shown. The point at which

boundary layer is separated from the object is clearly visible in the picture. The region of turbulent flow behind the object is called wake.

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Table 1: Drag of two-dimensional bodies at Re >10000

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Table 2: Drag of three-dimensional bodies at Re >10000

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Figure 11: Flow past a circular cylinder (a) laminar separation (b) turbulent separation. The point of separation, and

thus, the size of the wakes are different.

Recall that in flat plate boundary layer, we consider pressure gradient to be negligible, i.e., / 0p x∂ ∂ = , which is a valid assumption based on experimental observations. An argument to justify this assumption is:

Statement: The pressure in the boundary layer is dictated by the pressure in the inviscid flow (outside the boundary layer). In the inviscid flow, pressure changes according to Bernoulli’s equation,

2

constant2

p U zg gρ

Δ ΔΔ+ + =

Since in flat-plate flow zΔ is negligible, and streamlines are not curved, there is no change in pressure. Note that curved streamlines mean variation of velocity, and in an inviscid flow, based on the Bernoulli equation, when velocity change pressure should change too.

Unlike flat-plate flows, in flows around curved bodies, pressure changes! The proof for this statement is discussed in Chapter 8 of the textbook.

The theoretical “inviscid pressure” change on a cylinder can be obtained from the following relation

2

21 4sin1

2

pp pC

Uθ

ρ

∞−= = − (18)

where pC is called the dimensionless pressure coefficient, p is the pressure on the surface of the

cylinder, p∞ is the pressure of the free stream, U is the velocity of the free stream, and θ is the angle with respect to free stream velocity as shown in Fig. 11. In Fig. 12, the actual and theoretical pressure distributions on the boundary layer (or surface of the cylinder) are shown.

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Figure 12: Theoretical and actual pressure distributions in flow past a circular cylinder.

Effects of Pressure Gradient in Boundary Layers

Consider an inviscid flow around a cylinder with stream velocity of U and pressure of p∞ . Note that in an inviscid flow 0μ = and thus Re=∞ . Also, since viscosity is zero, there will be no shear (friction) on the surface of the cylinder and flow completely slides over the surface without any disturbance, see Fig. 13. In contrast, in viscous flows fluid particles experience energy loss due to friction and may not complete the path from A to F and separate in between.

Potential flow analysis (Chapter 8 of the textbook) which is appropriate for inviscid flows analysis, predicts the velocity and pressure at stagnation point (point A) and point C to be,

2A A

2C C

10 and232 and2

V p p U

V U p p U

ρ

ρ

∞

∞

= = +

= = − (19)

Figure 13: Inviscid flow around a cylinder.

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In Fig. 14, velocity and pressure variation on the surface of the cylinder is shown from point A to point F, i.e.,θ varies from 0 to 180 degrees. As mentioned in the above “Statement”, pressure distribution in a viscous boundary layer is dictated by the pressure distribution in the inviscid flow (if the boundary layer is sufficiently thin), hence we can say that if we have a viscous flow around a cylinder, then the pressure change in the boundary layer is very similar to what we discussed for the case of inviscid flow, as shown in Fig. 14.

Figure 14: Surface velocity and pressure distribution in an inviscid flow around a cylinder.

From point A to point C pressure decreases and / 0p x∂ ∂ < (where the curved flow path is shown by x). This negative pressure gradient is called “favorable pressure gradient”. Quite differently, from point C to F, pressure increases and / 0p x∂ ∂ > , called “adverse pressure gradient”. This means fluid particles have to compete against pressure and when their kinetic energy is not enough to overcome the pressure resistance, separation happens.