KUNDU Fluid Mechanics 2 Episode 4 pdf

are small compared to the speed of sound and in which the tcrnperature differences in the flow are small. This is discussed in Section 18. It is shown there that, undcr these restrictions, heating due to the viscous dissipation term is negligible in Eq. (4.66), and that the term -p(V u) can be combined with the left-hand side of Eq. (4.66) to give ([or a perfect gas) DT pc*- = -v -q. Dt If the hcat flux obeys the Fourier law Q = -kVT, then, if k = const., Eq. (4.67) simplifies to: (4.67) - KV~T. I %- (4.68) where K k/pC, is thc thermul dissivity, stated in m2/s and which is the same as that of the momentum diffusivity u. The viscous heating term t$ may be negligible in the thcrmal energy equa- tion (4.G6), but not in the mechanical energy cquation (4.62). In fact, there must be a sink of mechanical energy so that a steady state can be maintained in the prescnce of the various types of forcing. 15. Second IAW os Tlii?rmodynamic.s: Enhvpy Produclion The second law of thermodynamics esscntially says that real phenomena can only proceed in a direction in which the “disordcr” of an isolatcd system incrcases. Disor- der of a systcm is a measure of the degree of unifonnir?; of macroscopic properties in the system, which is the same as the dep of randomness in the molecular arrangc- nients that gcnerate thesc properties. In this conncction, disordcr, uniformity, and randomness havc essentially the same rncaning. For analogy, a tray containing rcd balls on one side and white balls on the othcr has more order than in an arrangement in which the balls arc mixed togcthcr. A real phenomenon must thereforc proceed in a direction in which such orderly arrangements dccrease because of “mixing.” Consider two possiblc states of an isolated fluid system, onc in which there are nonuniformities of temperaturc and velocity and the other in which thcse propertics are uniform. Both or these statcs have the same internal cnergy. Can the system spontaneously go from the state in which its properties are uniform to one in which they are nonuniform? The second law asserts that it cannot, based on cxperience. Natural proccsses, therefore, tend to causc mixing duc to transport of heat, momentum, and mass. A consequcnce of the swond law is that therc must exist aproperty called enrmpy, which is related to other thcrmodynamic propertics of the mcdium. In addition, thc second law says that the entropy of an isolated systcm can only increase; entropy is thercfore a measure of disordcr or randomness of a system. Lct S be the cntropy pcr unit mass. It is shown in Chapter 1, Scction 8 that the changc or entropy is related to 110 Cnnw&iMi Law the changes of internal energy e and specific volume u (= l/p) by P TdS = de + pdv =de - -dp. P2 Thc rate of change of cntropy following a fluid particle is therefore DS De p Dp Dt Dr p2 Dt' Tnserthg the internal energy equation (see Eq. (4.66)) T- = - - __ (4.69) De Dt p- = -v q - p(V u) + 4, and the continuity equation DP - = -p(V u), Dt the entropy production equation (4.69) becomes DS I aqi 4 P- = +- Dt T &ti T Using Fourier's law of heat conduction, this becomes The first term on the right-hand side, which has the form (heat gain)/T, is the cntropy gain due to reversible heat transfer because this term does not involve heat conduc- tivity. The last two terms, which are proportional to the square of temperature and velocity gradicnts, represcnt the entmpy production due to hcat conduction and vis- cous generation of heat. The second law of thermodynamics requks that the entropy production due to irreversible phenomena should be positive, so that An explicit appeal to the second law of thermodynamics is therefore not required in most analyscs of fluid flows bccause it has already been satisfied by laking positivc values for the molecular coeflicicnts of viscosity and thermal conductivity. If the flow is inviscid and nonheat: conducting, entropy is preservcd along the particle palhs. 16. BernouIIi hipalion Various conservation laws for mass, momentum, energy, and entropy wcre presented in the preceding sections. The well-known Bernoulli (4.46) equation is not a separate law, but is derived from the momentum equation for inviscid flows, namcly, the Euler equation (4.46): i)Ui aui il 1 aP - + u = (gz) - , at ’axj axi p ilxi where we have assumed that gravity g = -V(gz) is the only body force. The advective acceleration can be expressed in tern of vorticity as follows: whm we have used r;j = -&ijhOk (sce Eq. 3.23), and used the customary notation q2 = uz = twice kinetic encrgy. I Then the Euler equation becomes = (u x 0)i. (4.71) Now assume that p is a function of p only. A flow in which p = p(p) is called a barotmpic.fk,w, of which isothetmal and isentropic (p/pY = constant) flows arc special cascs. For such a flow we can writc (4.72) where dp/p is a perfect differential, and tbercfore the intcgral does not depend on the path of integration. To show this, note that whcre x is thc ‘‘field point:’ q is any arbitrary rcference point in the flow, and we have defined the following function of p alone: (4.74) Thc gradient of&. (4.73) gives ap dpap I ap where Eq. (4.74) has been used. The preceding equation is identical to Eq. (4.72). Using Q. (4.72), the Eulcr equation (4.71.) becomes Bui a [I 2 ST ] -+- -q + -++z =(uxo);. at axi 2 Defining the Bernoulli function I iq2 + 1 + gz = constant along streamlines and vortex lines 1 1 2 2 B = -42 + 1 + gz = -42 + P + gz, (4.78) thc Euler equation becomes (using vector notation) au at - + VB = u x 0. (4.75) (4.76) Bernouli equations are integrals of the conservation laws and have wide applicability as shown by the examples that follow. Important deductions can be made from the preceding equation by considering two special cases, namely a steady flow (rotational or irrotational) and an unsteady irrotational flow. These are described in what follows. Steady Flow In this case Eq. (4.76) reduces to VB=uxo. (4.77) The left-hand side is a vector normal to the surface B = constant, whereas the right-hand side is a vector perpendicular to both u and o (Figure 4.17). It follows that surfaces of constant B must contain the streamlines and vortex lines. Thus, an inviscid, steady, barotropic flow satisfies which is called Bemulli’s e4ualion. If, in addition, the flow is irrotational (o = 0), then Eq. (4.72) shows that ;q2 + 1 + gz = constant everywhere. vmx line (4.79) B = constant surface Figure 4.17 Bcrnoulli’z theorem. Note that the streamlines and vortex lincs can be at an arbitrary angle. P Fikwre 4.18 Flow over a solid objwl. Flow outside thc boundary layer is irrolalional. It may be shown that a sufficient condition for the existence of the surfaces con- taining streamlines and vortex lines is that the flow be barotropic. Tncidentally, thesc are called Lamb surfaces in honor of the distinguished English applied mathematician and hydrodynamicist, Horace Lamb. Tn a general, that is, nonbaroh-opjc Row, a path composed of streanilinc and vortex line segments can be drawn between any two points in a flow field. Thcn Eq. (4.78) is valid with the proviso that the integral be evaluated on the specific path chosen. As written, Eq. (4.78) requires the restTictions that the flow be stcady, inviscid, and have only gravity (or other conservative) body forces acting upon it. Tmtational flows are studied in Chapter 6. We shall note only the important.point here that, in a nonmtating frame of reference, bamtropic irrotational flows rcmain irrotational irviscous dTects are negligible. Considcr the flow around a solid object, say an airfoil (Figure 4.18). The flow is irrotational at all points outside the thin viscous layer closc to the surface of the body. This is bccause a particle P on a streamline outside the viscous layer started from some point S, where the flow is uniform and consequently irrotational. The Bernoulli equation (4.79) is therefore satisfied everywhere outsidc the viscous layer in this example. Unsteady Irrotational Flow An unsteady form oPBernoulli’s equation can be derived only if the flow is irrotational. For hotational flows thc velocity vector can be written as the gradient of a scalar potential Cp (called velocity potential): u = Vcp. (4.80) The validity of Eq. (4.80) can be checkcd by noting that it automatically satisfies the conditions of irrolationality aui auj axi axi - i#j. On inscrting Eq. (4.80) into Eq. (4.76), we obtain v [:: 7 + -42 ; + J $ + ,z] = 0, that is (4.81) 114 c7unmrryacionLuurn where the integrating function F(r) is independent of location. This form of the Bernoulli equation will be used in studying irrotational wave motions in Chapter 7. Energy Bernoulli Equation Return to Eq. (4.65) in the steady state with neither heat conduction nor viscous stresses. Then tij = -psii and Eq. (4.65) becomes If the body force per unit mass gi is conservative, say gravity, then +i = -(a/axi)(gz), which is the gradient of a scalar potential. In addition, from mass conservation, a(pui)/axi = 0 and thus (4.82) From Eq. (1.13). h = e + p/p. Eq. (4.82) now states that gradients of B’ = h + q2/2+gz must be normal to the local streamline direction ui. Then B’ = h +q2/2+ gz is a constant on streamlines. We showed in the previous section that inviscid, non-heat conducting flows are isentropic (S is conserved along particle paths), and in Eq. (1.1 8) we had the relation dp/p = dh when S = constant. Thus the path integral dp/p becomes a function h of the endpoints only if, in the momentum Bernoulli equation, both hcat conduction and viscous stresses may be neglected. This latter form hm the energy equation becomes very useful for high-speed gas flows to show the interplay between kinetic energy and internal energy or enthalpy or temperature along a streamline. 17. Applications of Bernoulli’s k$ualion Application of Bernoulli’s equation will now be illustrated for some simple flows. Pitot %be Consider first a simple device to measure the local velocity in a fluid stream by inserting a narrow bent tube (Figure 4.19). This is called apiror rube, after the French mathematician Henry Pitot (1 695-177 1 ), who used a bent glass tube to measure the velocity of the river Seine. Consider two points 1 and 2 at the same level, point 1 being away from the tube and point 2 being immediately in front of the open end where the fluid velocity is m. Friction is negligible along a streamline through 1 and 2, so that Bernoulli’s equation (4.78) gives from which the velocity is found to be itot tube P E’igure 4.19 Pilot tuhe for rncasuring vclocily in a duct. Prcssures at thc two points are found from thc hydrostatic balance PI = pghl and p2 = pgh2. so that [he velocity can bc found from Because it is assumcd that thc fluid density is very much greater than that of the atmosphcre to which the tubes are exposed, the pressures at the tops of the two fluid columns are assumed to be thc same. Thcy will actually differ by plumg(h2 - hl). Use of the hydrostatic approximation abovc station 1 is valid when the streamlines arc straight and parallel betwccn station 1 and thc upper wall. In working out this problem, the fluid dcnsity also has been laken to be a constant. Thc pressurc p2 measured by a pitot tubc is called “stagnation pressure:’ which is larger than the local static pressure. Evcn when there is no pitot tubc to meaqure thc stagnation pressure, it is customary to refcr LO the local valuc of thc quantity (p + pu2/2) as thc local stagnafiun pressure, defined as the pressure that would bc reached il he local flow is imgined to slow down to zcro velocity frictionlessly. The quanlity pu2/2 is somehcs called thc dynumic pm.wure; stagnation pressure is tbc sum of static and dynamic pressures. Orifice in a lhnk As another application or Bernoulli’s equalion, consider the flow though an orifice or opcning in a lank (Figure 4.20). The flow is slightly unsteady due to lowering 01 A A Distribution of (p -p,,J at orifice Figure 4.20 Flow through a sharp-edgcd orificc. Pressure has thc almosphcric value cvcrynherc mss seaion CC, its dishbution across orifice AA is indicated. the water level in the tank, but this effect is small if the tank area is large as compared to the orifice area. Viscous effects are negligible everywhere away from the walls of the tank. All streamlines can be traced back to the free surface in the tank, where they have the same value of thc Bernoulli constant B = y2/2 + p/p + gz. I1 .follows that the flow is irrotational, and B is constant throughout the flow. We want to apply Bernoulli’s equation between a point at the free surface in the tank and a point in the jet. However, the conditions right at the opening (section A in Figure 4.20) are not simple because the pressure is not uniform across the jet. Although pressure has the atmospheric value everywhere on the free surface of the jet (neglecting small surface tension effects), it is not equal to the atmospheric pressure inside the jet at this section. The streamlines at the orifice are curved, which requires that pressure must vary across the width of the jet in order to balance the centrifugal forcc. The pressure distribution across the orifice (section A) is shown in Figure 4.20. However, the streamlines in the jet become parallel at a short distance away from the orifice (section C in Figure 4.20), whcre the jet area is smaller than the orifice area. The pressure across section C is unifm and equal lo the atmospheric value because it has that value at the surface of the jet. Application of Bernoulli’s equation between a point on the free surface in the tank and a point at C gives from which the jet velocity is found as u = J2gh, Figure 4.21 Flow through a munded oriBcc. which simply states that the loss of potcnlial energy equals the gain of kinetic energy. The mass dow rate is rit = pA,u = PA&&, where A, is the area of the jet at C. For orifices having a sharp edge, A, has been round to bc %62% of thc orifice area. If the orifice happens to have a well-rounded opening (Figure 4.21), thcn he jet does not contract. The streamlines right at the exit are then parallel, and the pressure at the cxit is uniform and equal to the atmosphcric pressure. Consequently the mass flow rate is simply pAm, where A equals the orifice area. 18. Houwinesq Approximation For flows satisfying certain conditions, Boussinesq in 1903 suggested that the density changes in thc fluid can be neglected except in the gravity term where p is multiplicd by g. This approximation also treats the othcrpperties of the fluid (such asp, k, Cp) as constants. A formal jusNication, and the conditions under which the Boussinesq approximation holds, is givcn in Spiegel and Veronis (1960). Here we shall discuss the basis OF the approximation in a somewhat intuitive manner and examinc the resulting simplifications of the equations of motion. Continuity Equation The Boussinesq approximation replaces the continuity equation by the incompressible form v-u=o. (4.83) (4.84) However, this does not mcan that the density is regarded as constant along the direction of motion, but simply that the magnilude of p-’(Dp/Dt) is small in comparison to the magnitudes of the velocity gradients in V u. We can immediately think of several situations where the density variations cannot be neglected as such. The first situation is a steady flow with large Mach numbcrs (defined as U/c, where U is a typical measure of the flow speed and c is the speed of sound in the medium). At large Mach numbers the comprcssibility effects are large, because the large pressure changes cause large density changes. Jt is shown in Chapter 16 that compressibility effects are negligiblc in flows in which the Mach numbcr is <0.3. A typical value of c for air at ordinary temperatures is 350m/s, so that the assumption is good for speeds < 1.00 m/s. For water c = 1470 m/s, but the speeds normally achievable in liquids are much smaller than this value and therefore the incompressibility assumption is very good in liquids. A second situation in which the compressibility effects m impartant is unsteady flows. The wavcs would propagate at infinite speed if thc density variations are neglected. A third situation in which the compressibility effects are important occurs when the vertical scale of the flow is so large that the hydrostatic pressure variations cause large changes in density. In a hydrostatic field the vertical scale in which thc density changes become important is of order c2/g - 10 km for air. (This length agrees with the e-folding height RT/g of an “isothermal atmospherc,” because c2 = y RT; see Chapter 1, Section 10.) The Boussinesq approximation therefore requires that the vertical scale of the flow be L << c2/g. In the three situations mentioned the medium is regarded as “compressible,” in which the density depends strongly on pressure. Now suppose the compressibility effects are small, so that the density changes are caused by temperature changes alone, as in a thermal convection problem. .In this case the Boussinesq approximation applies whcn the temperature variations in the flow are small. Assume that p changes with T according to _- ” - -ar6T, P where a = -p-’(ap/aT), is the thermal expansion coefficient. Far a perfect gas a = 1 /T - 3 x K-l and for typical liquids a - 5 x I O4 K-’. With a temper- ature difference in Lhc fluid of 10 “C, thc varialion of density can be only a few percent a1 most. 1.t turns out that p-’(Dp/Df) can also bc no larger than a few percent of the velocity gradients in V u. To see this, assume that the flow field is characterized by a len@h scale L, a velocity scale U, and a tempcrature scale 61. By this we mean [...]... 987) Geophysical Fluid Dynamics, Ncw Y r Springer-Verlag ok Spiegel, E A and G Vcronis (1960) On thc Boussinesq approximation for a compmssible fluid Asfrophysical Journul131: 4 42 4 47 Stommcl H M and D W Moore (1989) An Introduction to the Corio1i.s Force New York: Columbia Univemity Press 'Itucsdcll, C A (19 52) Stokes' principle of viscosity JoumZ oJRationol Mechanics u d Analysi.s 1: 22 8 -23 1 Supplernim?alReading... (4. 83) by its incompressible form (4. 84) Momentum Equation Because of the incompressible continuity equation V u = 0, the stress tensor is givcn by Eq (4. 41) From Eq (4. 43, the equation of motion is then Du p= -vp Dt +p g +pv2u (4. 85) Consider a hypothetical static reference state in which the density is po everywhere and the pressure is po(t),so that Vpo = f i g Subtracting this state from Eq (4. 85)... 5 Vorticity Dynmics I Irifnniuchn 2 Ihrikx ruuI h e x luhs I 125 126 3 Hole of KwxMit!y in 1htdona.l and Irrntntioruil hrficw 126 Solid-Rodg HOUI~~OII 127 Irmtnrionalhrtex 127 Diecussion 130 4 Ke1nin:X (,'irculnhn 'I7mm-m 1 30 1)kcmsion of KelvZs 'I'hcni.c:rri 1 32 Helrii holm\irks 'I'hcorans 1 34 5 VorLki& Kqu&n i n ,Yonmlahg n I?mmf? 1 34 6 VorticiyEqualion in n Rotaling Rrmc... the integral form of energy conservation( 4 to a pillbox of infinites46) imal height 1 gives the rcsult niyi is continuous across thc interface, or explicity, k (aTl/an) = k2(aT2/an) at the interrace surface The hcat flux must be continuous l at the interfacc; it cannot store heat - Figure 4 .22 Interhccc bclwcen two mcdia; evaluation or boundary conditions  122 C n m m w hL u m Two more boundary conditions... Rotaling Rrmc 136 Memiing of (w V ) u 139 M ~ 1 1 of 2 ( 8 V)U 140 h ~ 7 h t m d o n o Vorhw 141 f 8 fi)rlcrL.%l 144 h'xmiws 145 I,ihmim ( ' k 146 ,id Supplemenhi1f i x d r g 147 1 Inhduciion Motion in circular streamlinesis called vortex motion The presence of closed streamlines does not necessarily mean that the fluid particles are rotating about their own centers, and we... (5 .21 )by &,**i( ):(, This gives (5 .22 ) The second k m on the left-hand side vanishes on noticing that enyiis antisymmetric r in q and i, whereas the derivative (u: /2 ll),iq is symmetric in q and i The third k m on the left-hand side of (5 .22 ) can be written as + =0 + ~1 [ P VxP Vpln, (5 . 24 ) which involvcsthe n-componentof the vector V p x V p The viscous term in Eq.(5 .22 ) can be written as = -V(&jSqk... / 2 and ur = 0 We can therefore apply the inviscid Eulcr equations, which in polar coordinates simplify to (5 .4) The pressure difference between two neighboring points is therefore d p = _dr aP dr + p d z = &mo2dr - pgdz, az where /.io = w r / 2 has been used Integration between any two points 1 and 2 gives pz - P I = $pwz(r ,2- r:> - pg(za - 21 ) (5.5) Surfaces of constant pressure are given by ~2 -... &~haqj)wk.jq = -vWk,nk -V&nqi&ijkWk,jq + vOn,jj = v%,jj (5 .25 ) If we use Eqs (5 .23 H5 .25 ), vorticity equation (5 .22 ) becomes awn - = un,j(wj at + 2Qj) - ujwn,j + ~1 [ P + V x P ~ l n vwn.jjV Changing the free index from n to i, this becomes In vector notation it is written as (5 .26 ) This is the vorticity equation for a nearly incompressible (that is, Boussinesq) fluid in rotating coordinates Here u and o are,... Figure 5.8 Coordinate system alignd with vorlicity vector - Meaning of 2( 8 V) u Orienting the z-axis along the direction of 8, this term becomes 2 ( 8 V)u = 2 2 (au/az) Suppressing all other terms in Eq (5 .26 ), we obtain C DO au - = 2C2Dt (barompic, inviscid, two-dimensional) as whose components are This shows that stretching of fluid lines in the z direction increases o,, whereas a tilting of vertical... interaction of two vortices of strengths rl and r2, with both rl and z 12 positive (that is, counterclockwisevorticity) Let h = h I hz be the distance betwccn the vortices (Figure 5.10) Then the velocity at point 2 due to vortex rl is directed upward, and equals + I 1’1 v,= 27 th’ Similarly, the velocity at point 1 due to vortcx r is downward, and equals 2 v2 r 2 = - 27 rh The vortex pair therefore rotates counterclockwisearound . (4. 74) Thc gradient of&. (4. 73) gives ap dpap I ap where Eq. (4. 74) has been used. The preceding equation is identical to Eq. (4. 72) . Using Q. (4. 72) , the Eulcr equation (4. 71.). 'Itucsdcll, C. A. (19 52) . Stokes' principle of viscosity. JoumZ oJRationol Mechanics ud Analysi.s 1: physical Journul131: 4 4 24 47. Univemity Press. 22 8 -23 1. Supplernim?al Reading. thus (4. 82) From Eq. (1.13). h = e + p/p. Eq. (4. 82) now states that gradients of B’ = h + q2 /2+ gz must be normal to the local streamline direction ui. Then B’ = h +q2 /2+