EXERCISES
3.6. REYNOLDS TRANSPORT THEOREM
of the ideal vortex flows described by (3.22) and (3.25). Near the center of rotation, a real vortex’s core flow is nearly solid-body rotation, but far from this core, real-vortex-induced flow is nearly irrotational. Two common idealizations of this behavior are the Rankine vortex defined by
uzðrÞ ¼
G=ps2¼const: forrs
0 forr>s
and uqðrÞ ¼
ðG=2ps2Þr forrs G=2pr forr>s
, (3.28) and the Gaussian vortex defined by
uzðrÞ ¼ G ps2exp
r2=s2
and uqðrÞ ¼ G 2pr
1exp
r2=s2
: (3.29)
In both cases,sis a core-size parameter that determines the radial distance where real vortex behavior transitions from solid-body rotation to irrotational vortex flow. For the Rankine vortex, this transition is abrupt and occurs at r¼s where uq reaches its maximum. For the Gaussian vortex, this transition is gradual and the maximum value ofuq is reached at rz1.12091s(see Exercise 3.26).
Consider a functionFthat depends on one independent spatial variable,x, and timet. In addition assume that the time derivative of its integral is of interest when the limits of integration, a and b, are themselves functions of time. Leibniz’s theorem states the time derivative of the integral ofF(x,t) betweenx¼a(t) andx¼b(t) is
d dt
Z x¼bðtÞ
x¼aðtÞ Fðx,tÞdx¼ Zb
a
vF
vtdx þdb
dtFðb,tÞ da
dtFða,tÞ, (3.30)
wherea,b,F, and their derivatives appearing on the right side of (3.30) are all evaluated at timet. This situation is depicted in Figure 3.17, where the three contributions are shown by dots and cross-hatches. The continuous line shows the integral!Fdx at timet, and the dashed line shows the integral at timet þdt. The first term on the right side of (3.30) is the integral ofvF/vtbetweenx¼aandb, the second term is the gain ofFat the upper limit which is moving at ratedb/dt, and the third term is the loss ofFat the lower limit which is moving at rateda/dt. The essential features of (3.30) are the total time derivative on the left, an integral over the partial time derivative of the integrand on the right, and terms that account for the time-dependence of the limits of integration on the right. These features persist when (3.30) is generalized to three dimensions.
A largely geometrical development of this generalization is presented here using notation drawn fromThompson (1972). Consider a moving volumeV*(t) having a (closed) surface A*(t) with outward normal n and let b denote the local velocity of A* (Figure 3.18). The volumeV* and its surfaceA* are commonly called acontrol volume and its control surface, respectively. The situation is quite general. The volume and its surface need not coincide with any particular boundary, interface, or surface. The velocity b need not be steady or uniform over A*(t). No specific coordinate system or origin of coordinates is needed. The goal of this effort is to determine the time derivative of the integral of a single-valued FIGURE 3.17 Graphical illustration of the Liebniz theorem. The three marked areas correspond to the three contributions shown on the right in (3.30). Hereda,db, andvF/vtare all shown as positive.
3. KINEMATICS
86
continuous functionF(x,t) in the volumeV*(t). The starting point for this effort is the defini- tion of a time derivative:
d dt
Z
VðtÞFðx,tÞdV ¼ lim
Dt/0
1 Dt
( Z
VðtþDtÞFðx,tþDtÞdV Z
VðtÞFðx,tÞdV )
: (3.31)
The geometry for the two integrals inside the {,}-braces is shown inFigure 3.18where solid lines are for timetwhile the dashed lines are for timetþDt. The time derivative of the inte- gral on the left is properly written as a total time derivative since the volume integration subsumes the possible spatial dependence ofF. The first term inside the {,}-braces can be expanded to four terms by defining the volume increment DVhVðtþDtÞ VðtÞ and Taylor expanding the integrand functionFðx,tþDtÞyFðx,tÞ þDtðvF=vtÞforDt/0:
Z
VðtþDtÞ
Fðx,tþDtÞdVy Z
VðtÞ
Fðx,tÞdVþ Z
VðtÞ
DtvFðx,tÞ vt dVþ
Z
DV
Fðx,tÞdV þ
Z
DV
DtvFðx,tÞ vt dV:
(3.32)
The first term on the right in (3.32) will cancel with the final term in (3.31), and, when the limit in (3.31) is taken, bothDt andDVgo to zero so the final term in (3.32) will not contribute because it is second order. Thus, when (3.32) is substituted into (3.31), the result is
d dt
Z
VðtÞFðx,tÞdV ¼ lim
Dt/0
1 Dt
( Z
VðtÞDtvFðx,tÞ vt dVþ
Z
DVFðx,tÞdV )
, (3.33)
and this limit may be taken once the relationship betweenDVandDtis known.
V*(t+Δt)
ndA
dA
V*(t)
A*(t) b
A*(t+Δt)
FIGURE 3.18 Geometrical depiction of a control volumeV*(t) having a surfaceA*(t) that moves at a nonuniform velocitybduring a small time incrementDt. WhenDtis small enough, the volume incrementDV¼V*(tþDt)eV*(t) will lie very nearA*(t), so the volume-increment element adjacent todAwill be (bDt),ndAwherenis the outward normal onA*(t).
3.6. REYNOLDS TRANSPORT THEOREM 87
To find this relationship consider the motion of the small area element dA shown in Figure 3.18. In timeDt,dAsweeps out an elemental volume (bDt),ndAof the volume incre- mentDV. Furthermore, this small element ofDVis located adjacent to the surfaceA*(t). All these elemental contributions toDVmay be summed together via a surface integral, and, asDtgoes to zero, the integrand value ofF(x,t) within these elemental volumes may be taken as that ofFon the surfaceA*(t), thus
Z
DVFðx,tÞdVy Z
AðtÞFðx,tÞðbDt,nÞdA as Dt/0: (3.34) Substituting (3.34) into (3.33), and taking the limit, produces the following statement of Rey- nolds transport theorem:
d dt
Z
VðtÞFðx,tÞdV¼ Z
VðtÞ
vFðx,tÞ vt dVþ
Z
AðtÞFðx,tÞb,ndA: (3.35) This final result follows the pattern set by Liebniz’s theorem that the total time derivative of an integral with time-dependent limits equals the integral of the partial time derivative of the integrand plus a term that accounts for the motion of the integration boundary. In (3.35), both inflows and outflows of F are accounted for through the dot product in the surface-integral term that monitors whetherA*(t) is locally advancing (b,n>0) or retreat- ing (b,n<0) alongn, so separate terms as in (3.30) are unnecessary. In addition, the (x,t)- space-time dependence of the control volume’s surface velocityb and unit normal n are not explicitly shown in (3.35) becauseband n are only defined onA*(t); neither is a field quantity likeF(x,t). Equation (3.35) is an entirely kinematic result, and it shows thatd/dt may be moved inside a volume integral and replaced by v/vtonly when the integration volume,V*(t), is fixed in space so that b¼0.
There are two physical interpretations of (3.35). The first, obtained when F¼1, is that volume is conserved as V*(t) moves through three-dimensional space, and under these conditions (3.35) is equivalent to (3.14) for small volumes (see Exercise 3.28). The second is that (3.35) is the extension of (3.5) to finite-size volumes (see Exercise 3.30). Nevertheless, (3.35) and judicious choices ofFandbare the starting points in the next chapter for deriving the field equations of fluid motion from the principles of mass, momentum, and energy conservation.
EXAMPLE 3.2
The base radiusrof a fixed-height right circular cone is increasing at the rate_r. Use Reynolds transport theorem to determine the rate at which the cone’s volume is increasing when the cone’s base radius isroif its height ish.
Solution
At any time, the volumeVof the right circular cone is:V¼1=3phr2, which can be differentiated directly and evaluated atr¼roto finddV=dt¼2=3phro_r. However, the task is to obtain this answer using (3.35). ChooseV* to perfectly enclose the cone so thatV*¼V, and setF¼1 in (3.35) so that the time derivative of the cone’s volume appears on the left:
3. KINEMATICS
88
dV=dt¼ Z
AðtÞb,ndA:
Use the cylindrical coordinate system shown inFigure 3.19with the cone’s apex at the origin. Here, b¼0 on the cone’s base whileb¼ ðz=hÞ_reRon its conical sides. The normal vector on the cone’s sides isn¼eRcosqezsinq where ro/h¼ tanq. Here, at the heightz, the cone’s surface area element isdA¼ztanqd4(dz/cosq), where4is the azimuthal angle, and the extra cosine factor enters because the conical surface is sloped. Thus, the volumetric rate of change becomes
dV dt ¼
Z h
z¼0
Z 2p 4¼0
z
h_reR,ðeRcosqezsinqÞz
? tanqd4
dz cosq
¼2prtan_ q h
Z h
z¼0z2dz ¼2
3ph2rtan_ q¼2 3phror,_ which recovers the answer obtained by direct differentiation.
EXERCISES
3.1. The gradient operator in Cartesian coordinates (x,y,z) is:
V¼exðv=vxÞ þeyðv=vyÞ þezðv=vzÞwhereex,ey, andezare the unit vectors. In cylindrical polar coordinates (R,4,z) having the same origin (seeFigure 3.3b), coordinates and unit vectors are related by:R¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
x2þy2
p ,4¼tan1ðy=xÞ, andz¼z;
andeR¼excos4þeysin4,e4¼ exsin4þeycos4, andez¼ez. Determine the following in the cylindrical polar coordinate system.
a) veR=v4andve4=v4 b) the gradient operatorV
c) the Laplacian operatorV,VhV2
h n b ro
θ z
R=ztanθ r•
FIGURE 3.19 Conical geometry for Example 3.2. The cone’s height is fixed but the radius of its circular surface (base) is increasing.
EXERCISES 89
d) the divergence of the velocity fieldV,u e) the advective acceleration term (u,V)u [See Appendix B for answers.]
3.2. Consider Cartesian coordinates (as given in Exercise 3.1) and spherical polar
coordinates (r,q,4) having the same origin (seeFigure 3.3c). Here coordinates and unit vectors are related by:r¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
x2þy2þz2
p ,q¼tan1ð ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2þy2
p =zÞ, and4¼tan1ðy=xÞ;
ander¼excos4sinqþeysin4sinqþezcosq,eq ¼excos4cosqþeysin4cosqezsinq, ande4¼ exsin4þeycos4. In the spherical polar coordinate system, determine the following items.
a) ver=vq,ver=v4,veq=vq,veq=v4, andve4=v4 b) the gradient operatorV
c) the LaplacianV,VhV2
d) the divergence of the velocity fieldV,u e) the advective acceleration term (u,V)u [See Appendix B for answers.]
3.3. Ifds¼(dx,dy,dz) is an element of arc length along a streamline (Figure 3.5) and u¼(u,v,w) is the local fluid velocity vector, show that ifdsis everywhere tangent tou thendx=u¼dy=v¼dz=w.
3.4. For the two-dimensional steady flow having velocity componentsu¼Syandv¼Sx, determine the following whenSis a positive real constant having units of inverse time.
a) equations for the streamlines with a sketch of the flow pattern b) the components of the strain-rate tensor
c) the components of the rotation tensor
d) the coordinate rotation that diagonalizes the strain-rate tensor, and the principal strain rates
3.5. Repeat Exercise 3.4 whenu¼eSxandv¼ þSy. How are the two flows related?
3.6. At the instant shown inFigure 3.2b, the (u,v)-velocity field in Cartesian coordinates is u¼Aðy2x2Þ=ðx2þy2Þ2, andv¼ 2Axy=ðx2þy2Þ2whereAis a positive constant.
Determine the equations for the streamlines by rearranging the first equality in (3.7) to readudyvdx¼0¼ ðvj=vyÞdyþ ðvj=vxÞdxand then looking for a solution in the formj(x,y)¼const.
3.7. Determine the equivalent of the first equality in (3.7) for two-dimensional (r,q)-polar coordinates, and then find the equation for the streamline that passes through (ro,qo) whenu¼(ur,uq)¼(A/r,B/r) whereAandBare constants.
3.8. Determine the streamline, path line, and streak line that pass through the origin of coordinates att¼t0whenu¼Uoþuxocos(ut) andv¼uxosin(ut) in two-dimensional Cartesian coordinates whereUois a constant horizontal velocity. Compare your results to those in Example 3.1 forUo/0.
3.9. Compute and compare the streamline, path line, and streak line that pass through (1,1,0) att¼0 for the following Cartesian velocity fieldu¼(x,eyt, 0).
3.10. Consider a time-dependent flow field in two-dimensional Cartesian coordinates where u¼[s=t2,v¼xy=[s, and[andsare constant length and time scales, respectively.
a) Use dimensional analysis to determine the functional form of the streamline throughx0at timet0.
3. KINEMATICS
90
b) Find the equation for the streamline throughx0at timet0and put your answer in dimensionless form.
c) Repeat part b) for the path line throughx0at timet0. d) Repeat part b) for the streak line throughx0at timet0.
3.11. The velocity components in an unsteady plane flow are given byu¼x=ð1þtÞand v¼2y=ð2þtÞ. Determine equations for the streamlines and path lines subject tox¼x0
att¼0.
3.12. Using the geometry and notation ofFigure 3.8, prove (3.9).
3.13. Determine the unsteady,vu/vt, and advective, (u,V)u, fluid acceleration terms for the following flow fields specified in Cartesian coordinates.
a) u¼ ðuðy,z,tÞ, 0, 0Þ
b) u¼UxwhereU¼ ð0, 0,UzðtÞÞ c) u¼AðtÞðx,y, 0Þ
d) u¼(Uoþuosin(kxeUt), 0, 0) whereUo,uo,k,andUare positive constants
3.14. Consider the following Cartesian velocity fieldu¼AðtÞðfðxÞ,gðyÞ,hðzÞÞwhereA,f,g, andhare nonconstant functions of only one independent variable.
a) Determinevu/vtand (u,V)uin terms ofA,f,g, andh, and their derivatives.
b) DetermineA,f,g, andhwhenDu/Dt¼0,u¼0 atx¼0, anduis finite fort>0.
c) For the conditions in part b), determine the equation for the path line that passes throughxoat timeto, and show directly that the accelerationaof the fluid particle that follows this path is zero.
3.15. If a velocity field is given by u¼ay andv¼0, compute the circulation around a circle of radiusrothat is centered on the origin. Check the result by using Stokes’
theorem.
3.16. Consider a plane Couette flow of a viscous fluid confined between two flat plates a distancebapart. At steady state the velocity distribution isu¼Uy/bandv¼w¼0, where the upper plate aty¼bis moving parallel to itself at speedU, and the lower plate is held stationary. Find the rates of linear strain, the rate of shear strain, and vorticity in this flow.
3.17. For the flow fieldu¼UþUx, whereUandUare constant linear- and angular- velocity vectors, use Cartesian coordinates to a) show thatSijis zero, and b) determineRij.
3.18. Starting with a small rectangular volume elementdV¼dx1dx2dx3, prove (3.14).
3.19. Let Oxyz be a stationary frame of reference, and let the z-axis be parallel with fluid vorticity vector in the vicinity of O so thatu¼Vu¼uzez in this frame of reference. Now consider a second rotating frame of reference Ox0y0z0 having the same origin that rotates about the z-axis at angular rate Uez. Starting from the kinematic relationship, u¼ ðUezÞ xþu0, show that in the vicinity of O the vorticity u0¼V0u0 in the rotating frame of reference can only be zero when 2U¼uz, where V0 is the gradient operator in the primed coordinates. The following unit vector transformation rules may be of use: e0x¼excosðUtÞ þ eysinðUtÞ, e0y¼ exsinðUtÞ þeycosðUtÞ, and e0z¼ez.
3.20. Consider a plane-polar area element having dimensionsdrandrdq. For two- dimensional flow in this plane, evaluate the right-hand side of Stokes’ theorem
EXERCISES 91
Ru,ndA¼R
u,dsand thereby show that the expression for vorticity in plane-polar coordinates is:uz¼1r vrvðruqÞ 1r vuvqr.
3.21. The velocity field of a certain flow is given byu¼2xy2þ2xz2,v¼x2y, andw¼x2z.
Consider the fluid region inside a spherical volumex2þy2þz2¼a2. Verify the validity of Gauss’ theoremRRR
V
V,udV¼RR
A
u,ndAby integrating over the sphere.
3.22. A flow field on thexy-plane has the velocity componentsu¼3xþyandv¼2x3y.
Show that the circulation around the circle (x1)2þ(y6)2¼4 is 4p.
3.23. Consider solid-body rotation about the origin in two dimensions:ur¼0 anduq¼u0r. Use a polar-coordinate element of dimensionrdqanddr, and verify that the circulation is vorticity times area. (InSection 3.5this was verified for a circular element
surrounding the origin.)
3.24. Consider the following steady Cartesian velocity fieldu¼
Ay
ðx2þy2Þb,ðxþ2þAxy2Þb, 0 . a) Determine the streamline that passes throughx¼ ðxo,yo, 0Þ.
b) ComputeRijfor this velocity field.
c) ForA>0, explain the sense of rotation (i.e., clockwise or counterclockwise) for fluid elements forb<1,b¼1, andb>1.
3.25. Using indicial notation (and no vector identities), show that the accelerationaof a fluid particle is given bya¼vu=vtþV1
2juj2
þuu, whereuis the vorticity.
3.26. Starting from (3.29), show that the maximumuqin a Gaussian vortex occurs when 1þ2ðr2=s2Þ ¼expðr2=s2Þ. Verify that this impliesrz1.12091s.
3.27. 1For the following time-dependent volumesV*(t) and smooth single-valued integrand functionsF, choose an appropriate coordinate system and show thatðd=dtÞR
VðtÞFdV obtained from (3.30) is equal to that obtained from (3.35).
a) V*(t)¼L1(t)L2L3is a rectangular solid defined by 0xiLi, whereL1depends on time whileL2andL3are constants, and the integrand functionF(x1,t) depends only on the first coordinate and time.
b) V*(t)¼(p/4)d2(t)Lis a cylinder defined by 0Rd(t)/2 and 0zL, where the cylinder’s diameterddepends on time while its lengthLis constant, and the integrand functionF(R,t) depends only on the distance from the cylinder’s axis and time.
c) V*(t)¼(p/6)D3(t) is a sphere defined by 0rD(t)/2 where the sphere’s diameter Ddepends on time, and the integrand functionF(r,t) depends only on the radial distance from the center of the sphere and time.
3.28. Starting from (3.35), setF¼1 and derive (3.14) whenb¼uandV*(t)¼dV/0.
3.29. For a smooth, single-valued functionF(x) that only depends on space and an
arbitrarily shaped control volume that moves with velocityb(t) that only depends on time, show thatðd=dtÞR
VðtÞFðxÞdV¼b,ðR
VðtÞVFðxÞdVÞ.
3.30. Show that (3.35) reduces to (3.5) whenV*(t)¼dV/0 and the control surface velocity bis equal to the fluid velocityu(x,t).
1Developed from Problem 1.9 on page 48 inThompson (1972) 3. KINEMATICS
92
Literature Cited
Meriam, J. L., and Kraige, L. G. (2007).Engineering Mechanics, Dynamics(6th ed). Hoboken, NJ: John Wiley and Sons, Inc.
Raffel, M., Willert, C., and Kompenhans, J. (1998).Particle Image Velocimetry. Berlin: Springer.
Riley, K. F., Hobson, M. P., and Bence, S. J. (1998).Mathematical Methods for Physics and Engineering (3rd ed.).
Cambridge, UK: Cambridge University Press.
Samimy, M., Breuer, K. S., Leal, L. G., and Steen, P. H. (2003).A Gallery of Fluid Motion. Cambridge, UK: Cambridge University Press.
Thompson, P. A. (1972).Compressible-Fluid Dynamics. New York: McGraw-Hill.
Van Dyke, M. (1982).An Album of Fluid Motion. Stanford, California: Parabolic Press.
Supplemental Reading
Aris, R. (1962).Vectors, Tensors, and the Basic Equations of Fluid Mechanics. Englewood Cliffs, NJ: Prentice-Hall. (The distinctions among streamlines, path lines, and streak lines in unsteady flows are explained, with examples.) Prandtl, L., and Tietjens, O. C. (1934).Fundamentals of Hydro- and Aeromechanics. New York: Dover Publications.
(Chapter V contains a simple but useful treatment of kinematics.)
Prandtl, L., and Tietjens, O. G. (1934).Applied Hydro- and Aeromechanics. New York: Dover Publications. (This volume contains classic photographs from Prandtl’s laboratory.)
SUPPLEMENTAL READING 93
C H A P T E R
4
Conservation Laws
O U T L I N E
4.1. Introduction 96
4.2. Conservation of Mass 96
4.3. Stream Functions 99
4.4. Conservation of Momentum 101 4.5. Constitutive Equation for
a Newtonian Fluid 111
4.6. Navier-Stokes Momentum
Equation 114
4.7. Noninertial Frame of Reference 116 4.8. Conservation of Energy 121
4.9. Special Forms of the Equations 125 4.10. Boundary Conditions 137 4.11. Dimensionless Forms of the
Equations and Dynamic
Similarity 143
Exercises 151
Literature Cited 168
Supplemental Reading 168
C H A P T E R O B J E C T I V E S
• To present a derivation of the governing equations for moving fluids starting from the principles of mass, momentum, and energy conservation for a material volume.
• To illustrate the application of the integral forms of the mass and momentum conservation equations to stationary, steadily moving, and accelerating control volumes.
• To develop the constitutive equation for a Newtonian fluid and provide the Navier- Stokes differential momentum equation.
• To show how the differential momentum equation is modified in noninertial frames of reference.
• To develop the differential energy equation and highlight its internal coupling between mechanical and thermal energies.
95
Fluid Mechanics, Fifth EditionDOI:10.1016/B978-0-12-382100-3.10004-6 Ó2012 Elsevier Inc. All rights reserved.
• To present several common extensions and simplified forms of the equations of motion.
• To derive and describe the dimensionless numbers that appear naturally when the equations of motion are put in dimensionless form.