EXERCISES
4.11. DIMENSIONLESS FORMS OF THE EQUATIONS AND DYNAMIC SIMILARITY
Multiply by the integrating factorz0and integrate. We obtain (rg/2s)z2 þ(1þ z02)1/2 ¼C.
EvaluateCasx/N,z/0,z0/0. ThenC¼1. We look atx¼0,z¼z(0)¼hto findh. The slope at the wall,z0¼tan(qþp/2)¼ cotq. Then 1þz02¼1þcot2q¼csc2q. Thus we now have (rg/2s)h2¼11/cscq¼1sinq, so thath2¼(2s/rg)(1sinq). Finally we seek to integrate to obtain the shape of the interface. Squaring and rearranging the result above, the differential equation we must solve may be written as 1þ(dz/dx)2¼[1(rg/2s)z2]2. Solving for the slope and taking the negative square root (since the slope is negative for positivex),
dz=dx ¼ n 1
1 rg=2s
z22o1=2 1
rg=2s z21
:
Define s/rg ¼ d2, z/d ¼ g. Rewriting the equation in terms of x/d andg, and separating variables:
2ð1g2=2Þg1
4g21=2 dg ¼ d
x=d :
The integrand on the left is simplified by partial fractions and the constant of integration is eval- uated atx¼0 whenh¼h/d. Finally,
cosh1ð2d=zÞ
4z2=d21=2
cosh1ð2d=hÞ þ ð4h2=d2Þ1=2 ¼ x=d gives the shape of the interface in terms ofx(z).
Analysis of surface tension effects results in the appearance of additional dimensionless parameters in which surface tension is compared with other effects such as viscous stresses, body forces such as gravity, and inertia. These are defined in the next section.
4.11. DIMENSIONLESS FORMS OF THE EQUATIONS AND
determined set the importance of the various terms in the governing differential equations, and thereby indicate which phenomena will be important in the resulting flow. This section describes and presents the primary dimensionless parameters or numbers required in the remainder of the text. Many others not mentioned here are defined and used in the broad realm of fluid mechanics.
The dimensionless parameters for any particular problem can be determined in two ways.
They can be deduced directly from the governing differential equations if these equations are known; this method is illustrated here. However, if the governing differential equations are unknown or the parameter of interest does not appear in the known equations, dimension- less parameters can be determined from dimensional analysis (see Section 1.11). The advan- tage of adopting the former strategy is that dimensionless parameters determined from the equations of motion are more readily interpreted and linked to the physical phenomena occurring in the flow. Thus, knowledge of the relevant dimensionless parameters frequently aids the solution process, especially when assumptions and approximations are necessary to reach a solution.
In addition, the dimensionless parameters obtained from the equations of fluid motion set the conditions under which scale model testing with small models will prove useful for pre- dicting the performance of larger devices. In particular, two flow fields are considered to be dynamically similar when their dimensionless parameters match, and their geometries are scale similar; that is, any length scale in the first flow field may be mapped to its counterpart in the second flow field by multiplication with a single scale ratio. When two flows are dynamically similar, analysis, simulations, or measurements from one flow field are directly applicable to the other when the scale ratio is accounted for. Moreover, use of standard dimensionless parameters typically reduces the parameters that must be varied in an exper- iment or calculation, and greatly facilitates the comparison of measured or computed results with prior work conducted under potentially different conditions.
To illustrate these advantages, consider the drag force FD on a sphere, of diameter dmoving at a speedUthrough a fluid of density rand viscosity m. Dimensional analysis (Section 1.11) using these five parameters produces the following possible dimensionless scaling laws:
FD
rU2d2 ¼ J rUd
m , or FDr m2 ¼ F
m
rUd : (4.99)
Both are valid, but the first is preferred because it contains dimensionless groups that either come from the equations of motion or are traditionally defined in the study of fluid dynamic drag. If dimensionless groups were not used, experiments would have to be conducted to determineFDas a function ofd, keepingU,r, andmfixed. Then, experiments would have to be conducted to determineFDas a function of U, keeping d,r, and mfixed, and so on.
However, such a duplication of effort is unnecessary when dimensionless groups are used.
In fact, use of the first dimensionless law above allows experimental results from a wide range of conditions to be simply plotted with two axes (seeFigure 4.21) even though the full complement of experiments may have involved variations in all five dimensional parameters.
The idea behind dimensional analysis is intimately associated with the concept of simi- larity. In fact, a collapse of all the data on a single graph such as the one inFigure 4.21is
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possible only because in this problem all flows having the same value of the dimensionless group known as the Reynolds number Re ¼rUd/m are dynamically similar. This dynamic similarity is assured because the Reynolds number appears when the equations of motion are cast in dimensionless form.
The use of dimensionless parameters pervades fluid mechanics to such a degree that this chapter and this text would be considered incomplete without this section, even though this topic is typically covered in first-course fluid mechanics texts. For clarity, the discussion begins with the momentum equation, and then proceeds to the continuity and energy equations.
Consider the flow of a fluid having nominal densityrand viscositymthrough a flow field characterized by a length scalel, a velocity scaleU, and a rotation or oscillation frequencyU.
The situation here is intended to be general so that the dimensional parameters obtained from this effort will be broadly applicable. Particular situations that would involve all five parameters include pulsating flow through a tube, flow past an undulating self-propelled body, or flow through a turbomachine.
The starting point is the Navier-Stokes momentum equation(4.39)simplified for incom- pressible flow. (The effect of compressibility is deduced from the continuity equation in the next subsection.)
r vu
vtþ ðu,VÞu ¼ VpþrgþmV2u (4.39b) FIGURE 4.21 Coefficient of dragCD for a sphere vs. the Reynolds number Re based on sphere diame- ter. At low Reynolds number CD ~ 1/Re, and above Re ~ 103,CD ~ constant (except for the dip between Re¼105 and 106). These behaviors (except for the dip) can be explained by simple dimensional reasoning.
The reason for the dip is the transition of the laminar boundary layer to a turbulent one, as explained in Chapter 9.
4.11. DIMENSIONLESS FORMS OF THE EQUATIONS AND DYNAMIC SIMILARITY 145
This equation can be rendered dimensionless by defining dimensionless variables:
xi ¼ xi=l, t ¼ Ut, uj ¼ uj=U, p ¼
ppN
rU2, and gj ¼ gj=g, (4.100) wheregis the acceleration of gravity. It is clear that the boundary conditions in terms of the dimensionless variables (4.100) are independent ofl, U, andU. For example, consider the viscous flow over a circular cylinder of radiusR. When the velocity scaleUis the free-stream velocity and the length scale is the radiusR, then, in terms of the dimensionless velocity u*¼u/Uand the dimensionless coordinater*¼r/R, the boundary condition at infinity is u*/1 asr*/N, and the condition at the surface of the cylinder isu*¼0 atr*¼1. In addi- tion, because pressure enters (4.39b) only as a gradient, the pressure itself is not of conse- quence; only pressure differences are important. The conventional practice is to render ppNdimensionless, wherepNis a suitably chosen reference pressure. Depending on the nature of the flow,ppNcould be made dimensionless in terms of a generic viscous stress mU/l, a hydrostatic pressurergl, or, as in (4.100), a dynamic pressurerU2. Substitution of (4.100) into (4.39) produces:
Ul U
vu
vt þ ðu,VÞu ¼ Vpþ gl
U2
gþ m
rUl
V2u, (4.101)
whereV¼ lV. It is apparent that two flows (having different values ofU,U,l,g, orm) will obey the same dimensionless differential equation if the values of the dimensionless groups Ul/U,gl/U2, and m/rUlare identical. Because the dimensionless boundary conditions are also identical in the two flows, it follows thatthey will have the same dimensionless solutions.
Products of these dimensionless groups appear as coefficients in front of different terms when the pressure is presumed to have alternative scalings (see Exercise 4.59).
The parameter groupings shown in [,]-brackets in (4.100) have the following names and interpretations:
St ¼ Strouhal numberhunsteady acceleration
advective accelerationf vu=vt
uðvu=vxÞfUU U2=l ¼ Ul
U, (4.102)
Re ¼ Reynolds numberhinertia force
viscous forcefruðvu=vxÞ m
v2u=vx2frU2=l mU=l2 ¼ rUl
m , and (4.103) Fr ¼ Froude numberh
inertia force gravity force
1=2 f
ruðvu=vxÞ rg
1=2 f
rU2=l rg
1=2
¼ U ffiffiffiffigl
p : (4.104) The Strouhal number sets the importance of unsteady fluid acceleration in flows with oscil- lations. It is relevant when flow unsteadiness arises naturally or because of an imposed frequency. The Reynolds number is the most common dimensionless number in fluid mechanics. Low Re flows involve small sizes, low speeds, and high kinematic viscosity such as bacteria swimming through mucous. High Re flows involve large sizes, high speeds, and low kinematic viscosity such as an ocean liner steaming at full speed.
St, Re, and Fr have to be equal for dynamic similarity of two flows in which unsteadiness and viscous and gravitational effects are important. Note that the mere presence of gravity does not make the gravitational effects dynamically important. For flow around an object
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in a homogeneous fluid, gravity is important only if surface waves are generated. Otherwise, the effect of gravity is simply to add a hydrostatic pressure to the entire system that can be combined with the pressure gradient (see “Neglect of Gravity in Constant Density Flows”
earlier in this chapter).
Interestingly, in a density-stratified fluid, gravity can play a significant role without the presence of a free surface. The effective gravity force in a two-layer situation is the buoyancy force per unit volume (r2r1)g, wherer1andr2are fluid densities in the two layers. In such a case, an internal Froude number is defined as:
Fr0h
inertia force buoyancy force
1=2 f
r1U2=l ðr2r1Þg
1=2
¼ Uffiffiffiffiffi g0l
p , (4.105)
whereg0 h g(r2r1)/r1is thereduced gravity. For a continuously stratified fluid having a maximum buoyancy frequencyN(see 1.29), the equivalent of (4.104) is Fr0hU=Nl. Alter- natively, the internal Froude number may be replaced by the Richardson Number¼Rih 1=Fr02 ¼ g0l=U2, which can also be refined to a gradient Richardson numberhN2ðzÞ=
ðdU=dzÞ2that is important in studies of instability and turbulence in stratified fluids.
Under dynamic similarity, all the dimensionless numbers match and there is one dimen- sionless solution. The dimensional consistency of the equations of motion ensures that all flow quantities may be set in dimensionless form. For example, the local pressure at point x¼(x,y,z) can be made dimensionless in the form
p x;t
pN 1=2
rU2 hCp ¼ J
St,Fr, Re;x l,Ut
, (4.106)
whereCp¼(ppN)/(1/2)rU2is called thepressure coefficient(or the Euler number), andJ represents the dimensionless solution for the pressure coefficient in terms of dimensionless parameters and variables. The factor of ½ in (4.106) is conventional but not necessary. Similar relations also hold for any other dimensionless flow variables such as velocityu/U. It follows that in dynamically similar flows, dimensionless flow variables are identical atcorresponding points and times(that is, for identical values ofx/l, andUt). Of course there are many instances where the flow geometry may require two or more length scales:l,l1,l2,.ln. When this is the case, the aspect ratiosl1/l,l2/l,.ln/lprovide a dimensionless description of the geometry, and would also appear as arguments of the functionJin a relationship like (4.106). Here a difference between relations (4.99) and (4.106) should be noted. Equation(4.99)is a relation betweenoverallflow parameters, whereas (4.106) holdslocallyat a point.
In the foregoing analysis we have assumed that the imposed unsteadiness in boundary conditions is important. However, time may also be made dimensionless viat*¼Ut/l, as would be appropriate for a flow with steady boundary conditions. In this case, the time derivative in (4.39) should still be retained because the resulting flow may still be naturally unsteady since flow oscillations can arise spontaneously even if the boundary conditions are steady. But, we know from dimensional considerations, such unsteadiness must have a time scale proportional tol/U.
In the foregoing analysis we have also assumed that an imposed velocityUis relevant.
Consider now a situation in which the imposed boundary conditions are purely unsteady.
4.11. DIMENSIONLESS FORMS OF THE EQUATIONS AND DYNAMIC SIMILARITY 147
To be specific, consider an object having a characteristic length scale l oscillating with a frequencyUin a fluid at rest at infinity. This is a problem having an imposed length scale and animposed time scale1/U. In such a case a velocity scaleU¼lUcan be constructed. The preceding analysis then goes through, leading to the conclusion that St¼1, Re¼Ul/n¼Ul2/n, and Fr¼U/(gl)1/2¼U(l/g)1/2have to be duplicated for dynamic similarity.
All dimensionless quantities are identical in dynamically similar flows. For flow around an immersed body, like a sphere, we can define the (dimensionless) drag and lift coefficients,
CDh FD 1=2
rU2A and CLh FL 1=2
rU2A, (4.107, 4.108)
whereAis a reference area, andFDandFLare the drag and lift forces, respectively, expe- rienced by the body; as in (4.106) the factor of 1/2 in (4.107) and (4.108) is conventional but not necessary. For blunt bodies such as spheres and cylinders, A is taken to be the maximum cross section perpendicular to the flow. Therefore, A¼ pd2/4 for a sphere of diameterd, andA¼bdfor a cylinder of diameterd and lengthb, with its axis perpendic- ular to the flow. For flows over flat plates, and airfoils, on the other hand,Ais taken to be theplanform area, that is,A¼sl; here, lis the average length of the plate or chord of the airfoil in the direction of flow and s is the width perpendicular to the flow, sometimes called thespan.
The values of the drag and lift coefficients are identical for dynamically similar flows. For flow about a steadily moving ship, the drag is caused both by gravitational and viscous effects so we must have a functional relation of the form CD ¼ CD(Fr, Re). However, in many flows gravitational effects are unimportant. An example is flow around a body that is far from a free surface and does not generate gravity waves. In this case, Fr is irrelevant, so CD¼CD(Re) is all that is needed when the effects of compressibility are unimportant.
This is the situation portrayed by the first member of (4.99) and illustrated inFigure 4.21.
A dimensionless form of the continuity equation should indicate when flow-induced pres- sure differences induce significant departures from incompressible flow. However, the simplest possible scaling fails to provide any insights because the continuity equation itself does not contain the pressure. Thus, a more fruitful starting point for determining the relative size ofV,uis (4.9),
V,u ¼ 1 r
Dr Dt ¼ 1
rc2 Dp
Dt, (4.9)
along with the assumption that pressure-induced density changes will be isentropic, dp¼c2drwherecis the sound speed (1.19). Using the following dimensionless variables,
xi ¼ xi=l, t ¼ Ut=l, uj ¼ uj=U, p ¼
ppN
roU2, and r ¼ r=ro, (4.109) whererois a reference density, the outside members of (4.9) can be rewritten:
V,u ¼ U2
c2 1
r Dp
Dt, (4.110)
which specifically shows that the square of
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M ¼ Mach numberh
inertia force compressibility force
1=2 f
rU2=l rc2=l
1=2
¼ U
c (4.111)
sets the size of isentropic departures from incompressible flow. In engineering practice, gas flows are considered incompressible whenM < 0.3, and from (4.110) this corresponds to
~10% departure from ideal incompressible behavior when (1/r*)(Dp*/Dt*) is unity. Of course, there may be nonisentropic changes in density too and these are considered in Thompson (1972, pp. 137e146). Flows in which M <1 are called subsonic, whereas flows in whichM > 1 are calledsupersonic. At high subsonic and supersonic speeds, matching Mach number between flows is required for dynamic similarity.
There are many possible thermal boundary conditions for the energy equation, so a fully general scaling of (4.60) is not possible. Instead, a simple scaling is provided here based on constant specific heats, neglect of my, and constant free-stream and wall temperatures, To
and Tw, respectively. In addition, for simplicity, an imposed flow oscillation frequency is not considered. The starting point of the scaling provided here is a mild revision of (4.60) that involves the enthalpyhper unit mass,
rDh Dt ¼ Dp
Dtþr3þ v vxi
kvT
vxi , (4.112)
where3is given by (4.58). UsingdhyCpdT, the following dimensionless variables:
3 ¼ rol23=moU2, m ¼ m=mo, k ¼ k=ko, T ¼ ðTToÞ=ðTwToÞ, (4.113) and those defined in (4.106), (4.107) becomes:
rDT Dt ¼
U2
Cp
TwTo
Dp
Dtþ
U2
Cp
TwTo
mo roUl
r3þ
ko
Cpmo mo roUl
VðkVTÞ: (4.114) Here the relevant dimensionless parameters are:
Ec ¼ Eckert numberh kinetic energy
thermal energy ¼ U2 Cp
TwTo
, (4.115)
Pr ¼ Prandtl numberhmomentum diffusivity thermal diffusivity ¼ n
k ¼ mo=ro
ko=roCp ¼ moCp
ko , (4.116) and we recognizeroUl/moas the Reynolds number in (4.114) as well. In low-speed flows, where the Eckert number is small the middle terms drop out of (4.114), and the full energy equation (4.112) reduces to (4.89). Thus, low Ec is needed for the Boussinesq approximation.
The Prandtl number is a ratio of two molecular transport properties. It is therefore a fluid property and independent of flow geometry. For air at ordinary temperatures and pressures, Pr¼0.72, which is close to the value of 0.67 predicted from a simplified kinetic theory model assuming hard spheres and monatomic molecules (Hirschfelder, Curtiss, & Bird, 1954, pp. 9e16). For water at 20C, Pr¼7.1. Dynamic similarity between flows involving thermal effects requires equality of the Eckert, Prandtl, and Reynolds numbers.
4.11. DIMENSIONLESS FORMS OF THE EQUATIONS AND DYNAMIC SIMILARITY 149
And finally, for flows involving surface tensions, there are several relevant dimensionless numbers:
We ¼ Weber numberh inertia force
surface tension forcefrU2l2
sl ¼ rU2l
s , (4.117)
Bo ¼ Bond numberh gravity force
surface tension forcefrl3g sl ¼ rl2g
s , (4.118)
Ca ¼ Capillary numberh viscous stress
surface tension stressfmU=l s=l ¼ mU
s : (4.119)
Here, for the Weber and Bond numbers, the ratio is constructed based on a ratio of forces as in (4.107) and (4.108), and not forces per unit volume as in (4.103), (4.104), and (4.111). At high Weber number, droplets and bubbles are easily deformed by fluid acceleration or decelera- tion, for example during impact with a solid surface. At high Bond numbers surface tension effects are relatively unimportant compared to gravity, as is the case for long-wavelength, ocean surface waves. At high capillary numbers viscous forces dominate those from surface tension; this is commonly the case in machinery lubrication flows. However, for slow bubbly flow through porous media or narrow tubes (low Ca) the opposite is true.
EXAMPLE 4.8
A ship 100 m long is expected to cruise at 10 m/s. It has a submerged surface of 300 m2. Find the model speed for a 1/25 scale model, neglecting frictional effects. The drag is measured to be 60 N when the model is tested in a towing tank at the model speed. Estimate the full scale drag when the skin-friction drag coefficient for the model is 0.003 and that for the full-scale prototype is half of that.
Solution
Estimate the model speed neglecting frictional effects. Then the nondimensional drag force depends only on the Froude number:
D=rU2l2 ¼ f U= ffiffiffiffi
p gl :
Equating Froude numbers for the model (denoted by subscript “m”) and full-size prototype (denoted by subscript “p”), we get
Um ¼ Up
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi gmlm=gplp
q ¼ 10 ffiffiffiffiffiffiffiffiffiffi p1=25
¼ 2m=s:
The total drag on the model was measured to be 60 N at this model speed. Of the total measured drag, a part was due to frictional effects. The frictional drag can be estimated by treating the surface of the hull as a flat plate, for which the drag coefficientCDis a function of the Reynolds number.
Using a value ofv¼106m2/s for water, we get
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Ul=n model
¼ 2
100=25
106 ¼ 8106, Ul=n
prototype
¼ 10 100
106 ¼ 109: The problem statement sets the frictional drag coefficients as
CDðmodelÞ ¼ 0:003;
CDðprototypeÞ ¼ 0:0015:
and these are consistent with Figure 9.11. Using a value of r ¼ 1000 kg/m3 for water, we estimate:
Frictional drag on model ¼ 1
2CDrU2A¼ ð0:5Þ ð0:003Þ ð1000Þ ð2Þ2ð300=252Þ ¼2:88N:
Out of the total model drag of60N,the wave drag is therefore602:88 ¼ 57:12N:
Now thewave dragstill obeys the scaling law above, which means thatD/rU2l2for the two flows are identical, whereDrepresents wave drag alone. Therefore,
Wave drag on prototype ¼ ðWave drag on modelÞ ðrp=rmÞ lp=lm
2 Up=Um
2
¼ 57:12ð1Þ ð25Þ2ð10=2Þ2¼ 8:92105N:
Having estimated the wave drag on the prototype, we proceed to determine its frictional drag.
Frictional drag on prototype ¼ 1
2CDrU2A¼ ð0:5Þ ð0:0015Þ ð1000Þ ð10Þ2ð300Þ ¼ 0:225105N:
Therefore, total drag on prototype¼(8.92þ0.225)105¼9.14105N.
If we did not correct for the frictional effects, and assumed that the measured model drag was all due to wave effects, then we would have found a prototype drag of
Dp ¼ Dmðrp=rmÞ lp=lm
2 Up=Um
2
¼ 60ð1Þ ð25Þ2ð10=2Þ2¼ 9:37105N:
EXERCISES
4.1. Let a one-dimensional velocity field beu¼u(x,t), withv¼0 andw¼0. The density varies asr¼r0(2cosut). Find an expression foru(x,t) ifu(0,t)¼U.
4.2. Consider the one-dimensional Cartesian velocity field:u ¼ ðax=t, 0, 0Þwhereais a constant. Find a spatially uniform, time-dependent density field,r¼r(t), that renders this flow field mass conserving whenr¼roatt¼to.
4.3. Find a nonzero density field r(x,y,z,t) that renders the following Cartesian velocity fields mass conserving. Comment on the physical significance and uniqueness of your solutions.
a) u ¼ ðUsinðUtkxÞ, 0, 0ÞwhereU,u,kare positive constants
[Hint: exchange the independent variablesx,tfor a single independent variable x¼utekx]
EXERCISES 151