Viscous Flow

T

he concept offluid viscosity was developed and defined in Chapter 7. Clearly, all fluids are viscous, but in certain situations and under certain conditions, afluid may be considered ideal or inviscid, making possible an analysis by the methods of Chapter 10.

Our task in this chapter is to consider viscousfluids and the role of viscosity as it affects theflow. Of particular interest is the case offlow past solid surfaces and the interrelations between the surfaces and theflowingfluid.

▶ 12.1

REYNOLDS’S EXPERIMENT

The existence of two distinct types of viscousflow is a universally accepted phenomenon.

The smoke emanating from a lighted cigarette is seen toflow smoothly and uniformly for a short distance from its source and then change abruptly into a very irregular, unstable pattern. Similar behavior may be observed for waterflowing slowly from a faucet.

The well-ordered type offlow occurs when adjacentfluid layers slide smoothly over one another with mixing between layers or lamina occurring only on a molecular level. It was for this type offlow that Newton’s viscosity relation was derived, and in order for us to measure the viscosity,m, thislaminarflow must exist.

The second flow regime, in which small packets of fluid particles are transferred between layers, giving it afluctuating nature, is called theturbulent flow regime.

The existence of laminar and turbulent flow, although recognized earlier, was first described quantitatively by Reynolds in 1883. His classic experiment is illustrated in Figure 12.1. Water was allowed to flow through a transparent pipe, as shown, at a rate controlled by a valve. A dye having the same specific gravity as water was introduced at the pipe opening and its pattern observed for progressively largerflow rates of water. At low rates of flow, the dye pattern was regular and formed a single line of color as shown in Figure 12.1(a). At highflow rates, however, the dye became dispersed throughout the pipe

(a) Re < 2300 (b) Re > 2300

Water Water

Dye Valve Dye Valve

Figure 12.1 Reynolds’s experiment.

154

cross section because of the very irregularfluid motion. The difference in the appearance of the dye streak was, of course, due to the orderly nature of laminarflow in thefirst case and to thefluctuating character of turbulentflow in the latter case.

The transition from laminar to turbulentflow in pipes is thus a function of thefluid velocity. Actually, Reynolds found thatfluid velocity was the only one variable determining the nature of pipeflow, the others being pipe diameter,fluid density, andfluid viscosity.

These four variables, combined into the single dimensionless parameter ReDrv

m (12-1)

form the Reynolds number, symbolized Re, in honor of Osborne Reynolds and his important contributions to fluid mechanics.

For flow in circular pipes, it is found that below a value for Reynolds number of 2300, theflow islaminar.Above this value theflow may be laminar as well, and indeed, laminarflow has been observed for Reynolds numbers as high as 40,000 in experiments wherein external disturbances were minimized. Above a Reynolds number of 2300, small disturbances will cause a transition to turbulentflow, whereas below this value distur- bances are damped out and laminarflow prevails. Thecritical Reynolds number for pipe flowthus is 2300.

▶ 12.2

DRAG

Reynolds’s experiment clearly demonstrated the two different regimes offlow: laminar and turbulent. Another manner of illustrating these differentflow regimes and their dependence upon Reynolds number is through the consideration of drag. A particularly illustrative case is that of externalflow (i.e.,flow around a body as opposed toflow inside a conduit).

The drag force due to friction is caused by the shear stresses at the surface of a solid object moving through a viscousfluid. Frictional drag is evaluated by using the expression

F

ACfrv²_∞

2 (12-2)

whereFis the force,Ais the area of contact between the solid body and thefluid,C_fis the coefficient of skin friction,ris thefluid density, andv_∞is the free-streamfluid velocity.

The coefficient of skin friction,C_f, which is defined by equation (12-2), is dimension- less.

The total drag on an object may be due to pressure as well as frictional effects. In such a situation another coefficient,C_D, is defined as

F

A_PCDrv²_∞

2 (12-3)

where F,r, andv_∞are as described above and, additionally, CDˆthe drag coefficient and AP ˆthe projected area of the surface

The value of A_Pused in expressing the drag for blunt bodies is normally the maximum projected area for the body.

12.2 Drag ◀ 155

The quantityrv²_∞=2 appearing in equations (12-2) and (12-3) is frequently called the dynamic pressure.

Pressure drag arises from two principal sources.¹One is induced drag, or drag due to lift. The other source is wake drag, which arises from the fact that the shear stress causes the streamlines to deviate from their inviscidflow paths, and in some cases to separate from the body altogether. This deviation in streamline pattern prevents the pressure over the rest of a body from reaching the level it would attain otherwise. As the pressure at the front of the body is now greater than that at the rear, a net rearward force develops.

In an incompressibleflow, the drag coefficient depends upon the Reynolds number and the geometry of a body. A simple geometric shape that illustrates the drag dependence upon the Reynolds number is the circular cylinder. The inviscid flow about a circular cylinder was examined in Chapter 10. The inviscid flow about a cylinder, of course, produced no drag, as there existed neither frictional nor pressure drag. The variation in the drag coefficient with the Reynolds number for a smooth cylinder is shown in Figure 12.2.

Theflow pattern about the cylinder is illustrated for several different values of Re. Theflow pattern and general shape of the curve suggest that the drag variation, and hence the effects of shear stress on theflow, may be subdivided into four regimes. The features of each regime will be examined.

Regime 1

In this regime the entireflow is laminar and the Reynolds number small, being less than 1.

Recalling the physical significance of the Reynolds number from Chapter 11 as the ratio of the inertia forces to the viscous forces, we may say that in regime 1 the viscous forces predominate. Theflow pattern in this case is almost symmetric, theflow adheres to the body, and the wake is free from oscillations. In this regime of the so-calledcreepingflow, viscous effects predominate and extend throughout theflowfield.

10^–1 10^{– 0} 10¹ 10² 10³ 10⁴ Regimes

III IV

II I

10⁵ 10⁶ CD

Reynolds number = 0.01

0.1 1 10 100

Figure 12.2 Drag coefficient for circular cylinders as a function of Reynolds number. Shaded regions indicate areas influenced by shear stress.

1A third source of pressure drag, wave drag, is associated with shock waves.

Regime 2

Two illustrations of theflow pattern are shown in the second regime. As the Reynolds number is increased, small eddies form at the rear stagnation point of the cylinder. At higher values of the Reynolds number, these eddies grow to the point at which they separate from the body and are swept downstream into the wake. The pattern of eddies shown in regime 2 is called a von Kármán vortex trail. This change in the character of the wake from a steady to an unsteady nature is accompanied by a change in the slope of the drag curve. The paramount features of this regime are (a) the unsteady nature of the wake and (b)flow separation from the body.

Regime 3

In the third regime the point offlow separation stabilizes at a point about 80 degrees from the forward stagnation point. The wake is no longer characterized by large eddies, although it remains unsteady. Theflow on the surface of the body from the stagnation point to the point of separation is laminar, and the shear stress in this interval is appreciable only in a thin layer near the body. The drag coefficient levels out at a near-constant value of approximately 1.

Regime 4

At a Reynolds number near 510⁵, the drag coefficient suddenly decreases to 0.3. When the flow about the body is examined, it is observed that the point of separation has moved past 90 degrees. In addition, the pressure distribution about the cylinder (shown in Figure 12.3) up to the point of separation is fairly close to the inviscidflow pressure distribution depicted in Figure 12.5. In thefigure it will be noticed that the pressure variation about the surface is a changing function of Reynolds number. The minimum point on the curves for Reynolds numbers of 10⁵and 610⁵are both at the point offlow separation. From thisfigure it is seen that separation occurs at a larger value ofq for Reˆ610⁵than it does for Reˆ10⁵.

θ, degrees

0 30 60 90 120 150 180

–4 –3 –2 –1 0 1 2

θ

Re = 6 × 10⁵ Inviscid flow

Re = 10⁵

Figure 12.3 Pressure distribution on a circular cylinder at various Reynolds numbers.

12.2 Drag ◀ 157

The layer of flow near the surface of the cylinder is turbulent in this regime, undergoing transition from laminar flow close to the forward stagnation point. The marked decrease in drag is due to the change in the point of separation. In general, a turbulentflow resistsflow separation better than a laminarflow. As the Reynolds number is large in this regime, it may be said that the inertial forces predominate over the viscous forces.

The four regimes offlow about a cylinder illustrate the decreasing realm of influence of viscous forces as the Reynolds number is increased. In regimes 3 and 4, the flow pattern over the forward part of the cylinder agrees well with the inviscidflow theory. For other geometries, a similar variation in the realm of influence of viscous forces is observed and, as might be expected, agreement with inviscid-flow predictions at a given Reynolds number increases as the slenderness of the body increases. The majority of cases of engineering interest involving externalflows haveflowfields similar to those of regimes 3 and 4.

Figure 12.4 shows the variation in the drag coefficient with the Reynolds number for a sphere, for infinite plates, and for circular disks and square plates. Note the similarity in form of the curve ofC_Dfor the sphere to that for a cylinder in Figure 12.2. Specifically, one may observe the same sharp decrease in C_D to a minimum value near a Reynolds number value of 510⁵. This is again due to the change from laminar to turbulentflow in the boundary layer.

Example 1

Evaluate the terminal velocity of a 7.5-mm-diameter glass sphere falling freely through (a) air at 300 K, (b) water at 300 K, and (c) glycerin at 300 K. The density of glass is 2250 kg/m³.

The terminal (steady-state) velocity of a falling body is reached when the force due tofluid drag matches the body’s weight.

In this case the weight of the glass sphere can be expressed as rspd³

6 g

0.1 0.2 0.5 1 2 5 10 20 50 200 1000 10,000 100,000 1,000,000 CD

0.02 0.05 0.1 0.2 0.5 1 2 5 10 20 50 100 200

Circular disks and square plates Sphere

Infinite plates Strokes drag

CD = 24/Re

D Infinite plates D Circular

disk and square plates

D Spheres

Figure 12.4 Drag coefficient versus Reynolds number for various objects.

Thefluid drag force is given by

CDrfv²_∞ 2

pd² 4 and a force balance yields

CDv²_∞ˆ4 3

rs

rfdg

The drag coefficient,CD, is plotted as a function of Reynolds number, Red, in Figure 12.4. AsCDis a function ofv_∞, we are unable to solve explicitly forv_∞unless Red<1, which would permit the use of Stokes’law in expressingCD.Atrial-and- error solution is, thus, required. The conditions to be satisfied are our force balance expression and the graphical relation betweenCDand Redin Figure 12.4.

For air at 300 K

v ˆ1:56910 ⁵m²=s r ˆ1:177 kg=m³ Red ˆdv_∞

v ˆ …7:510 ³m†v_∞ 1:56910 ⁵m²=s

ˆ478:0v_∞

(A)

Inserting known values into our force balance expression, we have CDv²_∞ˆ 4

3

2250 kg=m³

1:177 kg=m³…7:510 ³m†…9:81 m=s²†

ˆ187:5 m²=s²

(B) Normally the trial-and-error procedure to achieve a solution would be straightforward. In this case, however, the shape of theCDvs. Redcurve, given in Figure 12.4, poses a bit of a problem. Specifically, the value ofCDremains nearly uniform— that is, 0.4<CD<0.5—over a range in Redbetween 500<Red<10⁵: over three orders of magnitude!

In such a specific case, we will assumeCD≅0.4, and solve equation B forv_∞: v_∞ˆ 187:5

0:4 m²=s²

1=2

ˆ21:65 m=s Equation B then yields

Redˆ …478:0†…21:65† ˆ1:03510⁴

These results are compatible with Figure 12.4, although the absolute accuracy is obviously not great.

Finally, for air, we determine the terminal velocity to be, approximately,

v_∞≅21:6 m=s (a)

For water at 300 K,

v ˆ 0:87910 ⁶m²=s rˆ 996:1 kg=m³ Re_d ˆ …7:510 ³m†v_∞

0:87910 ⁶m²=sˆ8530v_∞ CDv²_∞ ˆ 4

3

2250 kg=m³

996 kg=m³ …7:510 ³m†…9:81 m=s²†

ˆ 0:2216 m²=s²

As in part (a), we will initially assumeCD≅0.4, and achieve the result that v_∞ˆ0:744 m=s Re_dˆ6350

12.2 Drag ◀ 159

These results, again, satisfy Figure 12.4. Thus, in water

v_∞ˆ0:744 m=s (b)

Finally, for glycerin at 300 K,

v ˆ 7:0810 ⁴m²=s rˆ 1260 kg=m³ Red ˆ …7:510 ³m†v_∞

7:0810 ⁴m²=sˆ10:59v_∞ CDv²_∞ ˆ 4

3

2250 kg=m³

1260 kg=m³…7:510 ³m†…9:81 m=s²†

ˆ 0:1752 m²=s²

In this case we suspect the Reynolds number will be quite small. As an initial guess we will assume Stokes’law applies, thus CDˆ24/Re.

Solving forv_∞for this case, we have

CDv²_∞ ˆ 24v

dv_∞v²_∞ˆ0:1752 m²=s² v_∞ ˆ…0:1752 m²=s²†…7:510 ³m†

24…7:0810 ⁴m²=s²†

ˆ0:0773 m=s

To validate the use of Stokes’law, we check the value of Reynolds number and get Redˆ…7:510 ³m†…0:0773 m=s†

7:0810 ⁴m²=s

ˆ0:819

which is in the allowable range. The terminal velocity in glycerin thus

v_∞ˆ0:0773 m=s (c)

▶ 12.3

THE BOUNDARY-LAYER CONCEPT

The observation of a decreasing region of influence of shear stress as the Reynolds number is increased led Ludwig Prandtl to the boundary-layer concept in 1904.

According to Prandtl’s hypothesis, the effects offluid friction at high Reynolds numbers are limited to a thin layer near the boundary of a body, hence the termboundary layer.

Further, there is no significant pressure change across the boundary layer. This means that the pressure in the boundary layer is the same as the pressure in the inviscidflow outside the boundary layer. The significance of the Prandtl theory lies in the simplification that it allows in the analytical treatment of viscousflows. The pressure, for example, may be obtained from experiment or inviscid flow theory. Thus, the only unknowns are the velocity components.

The boundary layer on a flat plate is shown in Figure 12.5. The thickness of the boundary layer, d, is arbitrarily taken as the distance away from the surface where

the velocity reaches 99% of the free-stream velocity. The thickness is exaggerated for clarity.

Figure 12.5 illustrates how the thickness of the boundary layer increases with distancex from the leading edge. At relatively small values of x, flow within the boundary layer is laminar, and this is designated as the laminar boundary-layer region. At larger values of xthe transition region is shown where fluctuations between laminar and turbulentflows occur within the boundary layer. Finally, for a certain value ofxand above, the boundary layer will always be turbulent. In the region in which the boundary layer is turbulent, there exists, as shown, a very thinfilm offluid called thelaminar sublayer, whereinflow is still laminar and large velocity gradients exist.

The criterion for the type of boundary layer present is the magnitude of Reynolds number, Re_x, known as thelocal Reynolds number, based on the distancexfrom the leading edge. The local Reynolds number is defined as

Re_xxvr

m (12-4)

For flow past a flat plate, as shown in Figure 12.5, experimental data indicate that for

(a) Rex<210⁵ the boundary layer is laminar

(b) 210⁵<Rex<310⁶ the boundary layer may be either laminar or turbulent (c) 310⁶<Rex the boundary layer is turbulent

▶ 12.4

THE BOUNDARY-LAYER EQUATIONS

The concept of a relatively thin boundary layer at high Reynolds numbers leads to some important simplifications of the Navier–Stokes equations. For incompressible, two-dimensionalflow over a flat plate, the Navier–Stokes equations are

r @vx

@t ‡vx@vx

@x ‡vy@vx

@y

ˆ@sxx

@x ‡@tyx

@y (12-5)

Figure 12.5 Boundary layer on aflat plate. (The thickness is exaggerated for clarity.)

12.4 The Boundary-Layer Equations ◀ 161

and

r @vy

@t ‡vx@vy

@x ‡vy@vy

@y

ˆ@txy

@x ‡@syy

@y (12-6)

where txyˆtyxˆm(@ vx/@y‡@ vy/@x) , sxxˆ Pˆ2m (@ vx/@x), and syyˆ P‡2m (@ vy/@y). The shear stress in a thin boundary layer is closely approximated bym(@ vx/@y).

This can be seen by considering the relative magnitudes of @vx/@y and @vy/@x From Figure 12.5, we may write vxj_d=vyj_d∼O…x=d†, where O signifies the order of magnitude. Then

@vx

@y ∼O vxj_d d

@vy

@x ∼O vyj_d x

so

@vx=@y

@vy=@x∼O x d 2

which, for a relatively thin boundary layer, is a large number, and thus@vx/@y

@

^{@ vy/@x.}

The normal stress at a large Reynolds number is closely approximated by the negative of the pressure as m…@vx=@x†∼O…mv_∞=x† ˆO…rv²_∞=Re_x†; therefore, sxx;syy; P.

When these simplifications in the stresses are incorporated, the equations for flow over aflat plate become

r @vx

@t ‡vx@vx

@x ‡vy@vx

@y

ˆ @P

@x‡m@²vx

@y² (12-7)

and

r @vy

@t ‡vx@vy

@x ‡vy@vy

@y

ˆ @P

@y‡m@²vy

@x² (12-8)

Furthermore,²the terms in the second equation are much smaller than those in thefirst equation, and thus @P/@y;0; hence @P/@xˆdP/dx, which according to Bernoulli’s equation is equal to rv_∞dv_∞/dx.

Thefinal form of equation (12-7) becomes

@vx

@t ‡vx@vx

@x ‡vy@vx

@y ˆv_∞dv_∞

dx ‡v@²vx

@y² (12-9)

The above equation, and the continuity equation

@vx

@x ‡@vy

@y ˆ0 (12-10)

are known as the boundary-layer equations.

2The order of magnitude of each term may be considered as above. For example, vx @vy

vx

∼Ov_∞^v_x^{∞ d}_x

ˆO ^v_x^2∞2d .

▶ 12.5

BLASIUS’S SOLUTION FOR THE LAMINAR BOUNDARY LAYER ON A FLAT PLATE

One very important case in which an analytical solution of the equations of motion has been achieved is that for the laminar boundary layer on a flat plate in steadyflow.

Forflow parallel to aflat surface,v_∞(x)ˆv_∞anddP/dxˆ0, according to the Bernoulli equation. The equations to be solved are now

vx@vx

@x‡vy@vx

@y ˆv@²vx

@y² (12-11a)

and @vx

@x ‡@vy

@y ˆ0 (12-11b)

with boundary conditionsvxˆvyˆ0 atyˆ0, andvxˆv_∞atyˆ

∞

^.

Blasius³obtained a solution to the set of equations (12-11) by first introducing the stream function, Y, as described in Chapter 10, which automatically satisfies the two- dimensional continuity equation, equation (12-11b). This set of equations may be reduced to a single ordinary differential equation by transforming the independent variablesx,y, toh and the dependent variables fromY(x,y) tof(h) where

h…x;y† ˆy 2

v_∞ vx 1=2

(12-12) and

f…h† ˆ Y…x;y†

…vxv_∞†¹⁼² (12-13)

The appropriate terms in equation (12-11a) may be determined from equations (12-12) and (12-13). The following expressions will result. The reader may wish to verify the mathematics involved.

vxˆ@Y

@yˆv_∞

2 f⁰…h† (12-14)

vyˆ @Y

@xˆ1 2

nv_∞ x 1=2

…hf⁰ f† (12-15)

@vx

@x ˆ v_∞h

4x f⁰⁰ (12-16)

@vx

@y ˆv_∞ 4

v_∞ vx 1=2

f⁰⁰ (12-17)

@²vx

@y² ˆv_∞ 8

v_∞

nxf⁰⁰⁰ (12-18)

Substitution of (12-14) through (12-18) into equation (12-11a) and cancellation gives, as a single ordinary differential equation,

f⁰⁰⁰‡f f⁰⁰ˆ0 (12-19)

3H. Blasius, Grenzshichten in Flüssigkeiten mit kleiner Reibung, ZMath. U. Phys. Sci.,1, 1908.

12.5 Blasius’s Solution for the Laminar Boundary Layer on a Flat Plate ◀ 163

with the appropriate boundary conditions

f ˆf⁰ˆ0 athˆ0 f⁰ˆ2 athˆ

∞

Observe that this differential equation, although ordinary, is nonlinear and that, of the end conditions on the variablef(h), two are initial values and the third is a boundary value.

This equation was solvedfirst by Blasius, using a series expansion to express the function, f(h), at the origin and an asymptotic solution to match the boundary condition ath ˆ

∞

^.

Howarth⁴later performed essentially the same work but obtained more accurate results.

Table 12.1 presents the significant numerical results of Howarth. A plot of these values is included in Figure 12.6.

A simpler way of solving equation (12-19) has been suggested in Goldstein,⁵who presented a scheme whereby the boundary conditions on the functionfare initial values.

If we define two new variables in terms of the constant,C, so that

fˆf=C (12-20)

and

xˆCh (12-21)

Table 12.1 Values off,f⁰,f⁰⁰, andvx=v_∞for laminarflow parallel to aflat plate (after Howarth) hˆ^y₂ ffiffiffiffi

n∞ nx

q f f⁰ f⁰⁰ _u^ux

∞

0 0 0 1.32824 0

0.2 0.0266 0.2655 1.3260 0.1328

0.4 0.1061 0.5294 1.3096 0.2647

0.6 0.2380 0.7876 1.2664 0.3938

0.8 0.4203 1.0336 1.1867 0.5168

1.0 0.6500 1.2596 1.0670 0.6298

1.2 0.9223 1.4580 0.9124 0.7290

1.4 1.2310 1.6230 0.7360 0.8115

1.6 1.5691 1.7522 0.5565 0.8761

1.8 1.9295 1.8466 0.3924 0.9233

2.0 2.3058 1.9110 0.2570 0.9555

2.2 2.6924 1.9518 0.1558 0.9759

2.4 3.0853 1.9756 0.0875 0.9878

2.6 3.4819 1.9885 0.0454 0.9943

2.8 3.8803 1.9950 0.0217 0.9915

3.0 4.2796 1.9980 0.0096 0.9990

3.2 4.6794 1.9992 0.0039 0.9996

3.4 5.0793 1.9998 0.0015 0.9999

3.6 5.4793 2.0000 0.0005 1.0000

3.8 5.8792 2.0000 0.0002 1.0000

4.0 6.2792 2.0000 0.0000 1.0000

5.0 8.2792 2.0000 0.0000 1.0000

4L. Howarth,“On the solution of the laminar boundary layer equations,”Proc. Roy. Soc. London,A164, 547 (1938).

5S. Goldstein,Modern Developments in Fluid Dynamics, Oxford University Press, London, 1938, p. 135.

then the terms in equation (12-19) become

f…h† ˆCf…x† (12-22)

f⁰ˆC²f⁰ (12-23)

f⁰⁰ˆC³f⁰⁰ (12-24)

and

f⁰⁰⁰ˆC⁴f⁰⁰⁰ (12-25)

The resulting differential equation inf(x) becomes

f⁰⁰⁰‡ff⁰⁰ˆ0 (12-26)

and the initial conditions onfare

fˆ0 f⁰ˆ0 f⁰⁰ˆ? atxˆ0 The other boundary condition may be expressed as follows:

f⁰…x† ˆf⁰…h†

C² ˆ 2

C² atxˆ

∞

An initial condition may be matched to this boundary condition if we letf⁰⁰(hˆ0) equal some constantA; thenf⁰⁰(xˆ0)ˆA/C³. The constantAmust have a certain value to satisfy the original boundary condition onf⁰. As an estimate we letf⁰⁰(xˆ0)ˆ2, givingAˆ2C³. Thus, initial values off, f⁰, andf⁰⁰are now specified. The estimate onf⁰⁰ (0) requires that

f⁰…

∞

^{† ˆ}_C²2 ˆ2 2 A

2=3

(12-27)

0 1.0 2.0 3.0

0 0.2 0.4 0.6 0.8 1.0

Blasius theory

Figure 12.6 Velocity distribution in the laminar boundary layer over aflat plate. Experimental data by J. Nikuradse (monograph, Zentrale F. wiss. Berichtswesen, Berlin, 1942) for the Reynolds number range from 1.0810⁵to 7.2810⁵.

12.5 Blasius’s Solution for the Laminar Boundary Layer on a Flat Plate ◀ 165