CHAPTER VIII Convective heat transfer The previous two chapters on issipative fluis were evote to flows ominate either by viscous effects (Chap. VI) or by convective motion (Chap. VII). In either case, the energy-conservation equation (III.36), an in particular the term representing heat conuction, was never playing a ominant role, with the exception of a brief mention of heat transport in the stuy of static Newtonian fluis (Sec. VI.1.1). The purpose of this Chapter is to shift the focus, an to iscuss motions of Newtonian fluis in which heat is transfere from one region of the flui to another. A first such type of transfer is heat conuction, which was alreay encountere in the static case. Uner the generic term convection, or convective heat transfer, one encompasses flows in which heat is also transporte by the moving flui, not only conuctively. Heat transfer will be cause by ifferences in temperature in a flui. Going back to the equations of motion, one can make a few assumptions so as to eliminate or at least suppress other effects, an emphasize the role of temperature graients in moving fluis (Sec VIII.1). A specific instance of flui motion riven by a temperature ifference, yet also controlle by the flui viscosity, which allows for a richer phenomenology, is then presente in Sec. VIII.2. VIII.1 Equations of convective heat transfer The funamental equations of the ynamics of Newtonian fluis seen in Chap. III inclue heat conuction, in the form of a term involving the graient of temperature, yet the change in time of temperature oes not explicitly appear. To obtain an equation involving the time erivative of temperature, some rewriting of the basic equations is thus neee, which will be one together with a few simplifications (Sec. VIII.1.1). Conuction in a static flui is then recovere as a limiting case. In many instances, the main effect of temperature ifferences is however rather to lea to variations of the mass ensity, which in turn trigger the flui motion. To have a more aapte escription of such phenomena, a few extra simplifying assumptions are mae, leaing to a new, close set of couple equations (Sec. VIII.1.2). VIII.1.1 Basic equations of heat transfer Consier a Newtonian flui submitte to conservative volume forces f ~ V = r ~.Itsmotionis governe by the laws establishe in Chap. III, namely by the continuity equation, the Navier Stokes equation, an the energy-conservation equation or equivalently the entropy-balance equation, which we now recall. Expaning the ivergence of the mass flux ensity, the continuity equation (III.9) becomes D (t,~r) Dt = (t,~r) ~ r ~v(t,~r). (VIII.1a) In turn, the Navier Stokes equation (III.30a) may be written in the form (t,~r) D~v(t,~r) Dt = ~ rp (t,~r) (t,~r) ~ r (t,~r)+2 ~ r (t,~r)s(t,~r) + ~ r (t,~r) ~ r ~v(t,~r). (VIII.1b)
120 Convective heat transfer Eventually, straightforwar algebra using the continuity equation allows one to rewrite the entropy balance equation (III.41b) as (t,~r) D apple s(t,~r) = 1 r apple(t,~r) Dt (t,~r) T (t,~r) ~ rt ~ (t,~r) + 2 (t,~r) T (t,~r) (t,~r) 2. S(t,~r): S(t,~r)+ ~r ~v(t,~r) T (t,~r) (VIII.1c) Since we wish to isolate effects irectly relate with the transfer of heat, or playing a role in it, we shall make a few assumptions, so as to simplify the above set of equations. The transport coefficients,, apple epen on the local thermoynamic state of the flui, i.e. on its local mass ensity an temperature T, an thereby inirectly on time an position. Nevertheless, they will be taken as constant an uniform throughout the flui, an taken out of the various erivatives in Eqs. (VIII.1b) (VIII.1c). This is a reasonable assumption as long as only small variations of the flui properties are consiere, which is consistent with the next assumption. Somewhat abusively, we shall in fact even allow ourselves to consier resp. apple as uniform in Eq. (VIII.1b) resp. (VIII.1c), later replace them by relate (iffusion) coefficients = / resp. = apple/ c P,anthenconsierthelatterasuniformconstantquantities. The whole proceure is only justifie in that one can check by comparing calculations using this assumption with numerical computations performe without the simplifications that it oes not lea to omitting a physical phenomenon. The flui motions uner consieration will be assume to be slow, i.e. to involve a small flow velocity, in the following sense: The incompressibility conition ~ r ~v(t,~r) =0will hol on the right han sies of each of Eqs. (VIII.1). Accoringly, Eq. (VIII.1a) simplifies to D (t,~r)/dt =0while Eq. (VIII.1b) becomes the incompressible Navier Stokes equation @~v(t,~r) @t + ~v(t,~r) ~r ~v(t,~r) = in which the kinematic viscosity is taken to be constant. 1 (t,~r) ~ rp (t,~r) ~ r (t,~r)+ 4~v(t,~r), (VIII.2) The rate of shear is small, so that its square can be neglecte in Eq. (VIII.1c). Accoringly, that equation simplifies to (t,~r)t (t,~r) D apple s(t,~r) = apple4t (t,~r). (VIII.3) Dt (t,~r) The term on the left han sie of that equation can be further rewritten. As a matter of fact, one can show that the ifferential of the specific entropy in a flui particle is relate to the change in temperature by s T = c P T, (VIII.4) where c P enotes the specific heat capacity at constant pressure of the flui. In turn, this relation translates into an ientity relating the material erivatives when the flui particles are followe in their motion. The left member of Eq. (VIII.3) may then be expresse in terms of the substantial erivative of the temperature. Introucing the thermal iffusivity (lxxii) apple c P, which will from now on be assume to be constant an uniform in the flui, where c P (lxxii) Temperaturleitfähigkeit (VIII.5) is the
VIII.1 Equations of convective heat transfer 121 volumetric heat capacity at constant pressure, one eventually obtains DT (t,~r) Dt = @T(t,~r) @t + ~v(t,~r) ~r T (t,~r) = 4T (t,~r), (VIII.6) which is sometimes referre to as (convective) heat transfer equation. If the flui is at rest or if its velocity is small enough to ensure that the convective term~v ~rt remains negligible, Eq. (VIII.6) simplifies to the classical heat iffusion equation, with iffusion constant. The thermal iffusivity thus measures the ability of a meium to transfer heat iffusively, just like the kinematic shear viscosity characterizes the iffusive transfer of momentum. Accoringly, both coefficients have the same imension L 2 T 1, an can thus be meaningfully compare. Their relative strength is measure by the imensionless Prantl number Pr = c P apple (VIII.7) which in contrast to the Mach, Reynols, Froue, Ekman, Rossby... numbers encountere in the previous chapters is entirely etermine by the flui, inepenent of any flow characteristics. VIII.1.2 Boussinesq approximation If there is a temperature graient in a flui, it will lea to a heat flux ensity, an thereby to the transfer of heat, thus influencing the flui motion. However, heat exchanges by conuction are often slow except in metals, so that another effect ue to temperature ifferences is often the first to play a significant role, namely thermal expansion (or contraction), which will lea to buoyancy (Sec. IV.1.1) when a flui particle acquires a mass ensity ifferent from that of its surrounings. The simplest approach to account for this effect, ue to Boussinesq, (52) consists in consiering that even though the flui mass ensity changes, nevertheless the motion can to a very goo approximation be viewe as incompressible which is what was assume in Sec. VIII.1.1: ~r ~v(t,~r) ' 0, (VIII.8) where ' is use to allow for small relative variations in the mass ensity, which are irectly relate to the expansion rate through Eq. (VIII.1a). Denoting by T 0 a typical temperature in the flui an 0 the corresponing mass ensity (strictly speaking, at a given pressure), the effect of thermal expansion on the latter reas with ( ) = 0 (1 (V ) ), (VIII.9) T T 0 (VIII.10) the temperature ifference measure with respect to the reference value, an (V ) 1 @ @T P,N (VIII.11) the thermal expansion coefficient for volume, where the erivative is taken at the thermoynamic point corresponing to the reference value 0. Strictly speaking, the linear regime (VIII.9) (53) only hols when (V ) 1, which will be assume hereafter. (52) Hence its enomination Boussinesq approximation (for buoyancy). (53)... which is obviously the beginning of a Taylor expansion.
122 Convective heat transfer Consistent with relation (VIII.9), the pressure term in the incompressible Navier Stokes equation can be approximate as 1 (t,~r) ~ rp (t,~r) ' ~ rp (t,~r) 0 1+ (V ) (t,~r). Introucing an effective pressure P e which accounts for the leaing effect of the potential from which the volume forces erive, P e. (t,~r) P (t,~r)+ 0 (t,~r), one fins 1 rp (t,~r) ~ (t,~r) r ~ rp ~ e. (t,~r) (t,~r) ' + (V ) (t,~r) r ~ (t,~r), 0 where a term of subleaing orer (V ) rp ~ e. has been roppe. To this level of approximation, the incompressible Navier Stokes equation (VIII.2) becomes @~v(t,~r) @t + ~v(t,~r) ~r ~v(t,~r) = ~ rp e. (t,~r) 0 + (V ) (t,~r) ~ r (t,~r)+ 4~v(t,~r). (VIII.12) This form of the Navier Stokes equation emphasizes the role of a finite temperature ifference in proviing an extra force ensity which contributes to the buoyancy, supplementing the effective pressure term. Eventually, efinition (VIII.10) together with the convective heat transfer equation (VIII.6) lea at once to @ (t,~r) + ~v(t,~r) @t ~r (t,~r) = 4 (t,~r). (VIII.13) The (Oberbeck (ap) )Boussinesq equations (VIII.8), (VIII.12), an (VIII.13) represent a close system of five couple scalar equations for the ynamical fiels~v, which in turn yiels the whole variation of the mass ensity an P e.. VIII.2 Rayleigh Bénar convection A relatively simple example of flow in which thermal effects play a major role is that of a flui between two horizontal plates at constant but ifferent temperatures, the lower plate being at the higher temperature, in a uniform gravitational potential r ~ (t,~r) =~g, in the absence of horizontal pressure graient. The istance between the two plates will be enote by, an the temperature ifference between them by T, where T>0when the lower plate is warmer. When neee, a system of Cartesian coorinates will be use, with the (x, y)-plane miway between the plates an a vertical z-axis, with the acceleration of gravity pointing towars negative values of z, i.e. (t,~r) =gz. VIII.2.1 Phenomenology of the Rayleigh Bénar convection VIII.2.1 a :::::::: :::::::::::::::::::::::: Experimental finings If both plates are at the same temperature or if the upper one is the warmer ( T<0), the flui between them can simply be at rest, with a stationary linear temperature profile. As a matter of fact, enoting by T 0 resp. P 0 the temperature resp. pressure at a point at z =0 an 0 the corresponing mass ensity, one easily checks that equations (VIII.8), (VIII.12), (VIII.13) amit the static solution ~v st. (t,~r) =~0, st. (t,~r) = z T, P e.,st.(t,~r) =P 0 0 g z2 2 (V ) T, (VIII.14) (ap) A. Oberbeck, 1849 1900
VIII.2 Rayleigh Bénar convection 123 with the pressure given by P st. (t,~r) =P e.,st. (t,~r) 0 gz. Since z/ < 1 2 an (V ) T 1, one sees that the main part of the pressure variation ue to gravity is alreay absorbe in the efinition of the effective pressure. If T =0, one recognizes the usual linear pressure profile of a static flui at constant temperature in a uniform gravity fiel. One can check that the flui state efine by the profile (VIII.14) is stable against small perturbations of any of the ynamical fiels. To account for that property, that state (for a given temperature ifference T ) will be referre to as equilibrium state. Increasing now the temperature of the lower plate with respect to that of the upper plate, for small positive temperature ifferences T nothing happens, an the static solution (VIII.14) still hols an is still stable. When T reaches a critical value T c, the flui starts eveloping a pattern of somewhat regular cylinrical omains rotating aroun their longituinal, horizontal axes, two neighboring regions rotating in opposite irections. These omains in which warmer an thus less ense flui rises on the one sie while coler, enser flui escens on the other sie, are calle Bénar cells. (aq) 6 Figure VIII.1 Schematic representation of Bénar cells between two horizontal plates. The transition from a situation in which the static flui is a stable state, to that in which motion evelops i.e. the static case is no longer stable, is referre to as (onset of the) Rayleigh Bénar instability. Since the motion of the flui appears spontaneously, without the nee to impose any external pressure graient, it is an instance of free convection or natural convection in opposition to force convection). Remarks: Such convection cells are omnipresent in Nature, as e.g. in the Earth mantle, in the Earth atmosphere, or in the Sun convective zone. When T further increases, the structure of the convection pattern becomes more complicate, eventually becoming chaotic. In a series of experiments with liqui helium or mercury, A. Libchaber (ar) an his collaborators observe the following features [36, 37, 38]: Shortly above T c,thestablefluistateinvolve cylinrical convective cells with a constant profile. Above a secon threshol, oscillatory convection evelops: that is, unulatory waves start to propagate along the surface of the convective cells, at first at a unique (angular) frequency! 1,then as T further increases also at higher harmonics n 1! 1, n 1 2 N. Asthetemperatureifference T reaches a thir threshol, a secon unulation frequency! 2 appears, incommensurate with! 1,lateraccompaniebythecombinations n 1! 1 + n 2! 2, with n 1,n 2 2 N. Athigher T, the oscillator with frequency! 2 experiences a shift from its proper frequency to a neighboring submultiple of! 1 e.g.,! 1 /2 in the experiments with He, illustrating the phenomenon of frequency locking. Forevenhigher T, submultiplesof! 1 appear ( frequency emultiplication ), then a low-frequency continuum, an eventually chaos.? (aq) H. Bénar, 1874 1939. (ar) A. Libchaber, born1934
124 Convective heat transfer Each appearance of a new frequency may be seen as a bifurcation. The ratios of the experimentally measure lengths of consecutive intervals between successive bifurcations provie an estimate of the (first) Feigenbaum constant (as) in agreement with its theoretical value thereby proviing the first empirical confirmation of Feigenbaum s theory. VIII.2.1 b ::::::::: ::::::::::::::::::::::: Qualitative iscussion Consier the flui in its equilibrium state of rest, in the presence of a positive temperature ifference T, so that the lower layers of the flui are warmer than the upper ones. If a flui particle at altitue z acquires, for some reason, a temperature that iffers from the equilibrium temperature measure with respect to some reference value (z), then its mass ensity given by Eq. (VIII.9) will iffer from that of its environment. As a result, the Archimees force acting on it no longer exactly balances its weight, so that it will experience a buoyancy force. For instance, if the flui particle is warmer that its surrounings, it will be less ense an experience a force irecte upwars. Consequently, the flui particle shoul start to move in that irection, in which case it encounters flui which is even coler an enser, resulting in an increase buoyancy an a continue motion. Accoring to that reasoning, any vertical temperature graient shoul result in a convective motion. There are however two effects that counteract the action of buoyancy, an explain why the Rayleigh Bénar instability necessitates a temperature ifference larger than a given threshol. First, the rising particle flui will also experience a viscous friction force from the other flui regions it passes through, which slows its motion. Seconly, if the rise of the particle is too slow, heat has time to iffuse by heat conuction through its surface: this tens to equilibrate the temperature of the flui particle with that of its surrounings, thereby suppressing the buoyancy. Accoringly, we can expect to fin that the Rayleigh Bénar instability will be facilitate when (V ) Tg i.e. the buoyancy per unit mass increases, as well as when the thermal iffusivity an the shear viscosity ecrease. Translating the previous argumentation in formulas, let us consier a spherical flui particle with raius R, an assume that it has some vertically irecte velocity v, while its temperature initially equals that of its surrounings. With the flui particle surface area, proportional to R 2, an the thermal iffusivity apple, one can estimate the characteristic time for heat exchanges between the particle an the neighboring flui, namely Q = C R2 with C a geometrical factor. If the flui particle moves with constant velocity v uring that uration Q, while staying at almost constant temperature since heat exchanges remain limite, the temperature ifference it acquires with respect to the neighboring flui is = @ @z z = @ @z v Q = C T R 2 v, where T/ is the temperature graient in the flui impose by the two plates. This temperature ifference gives rise to a mass ensity ifference = 0 (V ) = C 0 v R2 between the particle an its surrounings. As a result, flui particle experiences an upwars irecte buoyancy 4 3 R3 g = 4 C 3 0gv R5 (V ) T. (VIII.15) (as) M. Feigenbaum, born1944 (V ) T,
VIII.2 Rayleigh Bénar convection 125 On the other han, the flui particle is slowe in its vertical motion by the ownwars oriente Stokes friction force acting on it, namely, in projection on the z-axis F Stokes = 6 R v. (VIII.16) Note that assuming that the velocity v remains constant, with a counteracting Stokes force that is automatically the goo one, relies on the picture that viscous effects aapt instantaneously, i.e. that momentum iffusion is fast. That is, the above reasoning actually assumes that the Prantl number (VIII.7) is much larger than 1; yet its result is inepenent from that assumption. Comparing Eqs. (VIII.15) an (VIII.16), buoyancy will overcome friction, an thus the Rayleigh Bénar instability take place, when 4 C 3 0gv R5 (V ) T > 6 R 0 v, (V ) TgR 4 > 9 2C. Note that the velocity v which was invoke in the reasoning actually rops out from this conition. Taking for instance R = /2 which maximizes the left member of the inequality, this becomes Ra (V ) Tg 3 > 72 C =Ra c. Ra is the so-calle Rayleigh number an Ra c its critical value, above which the static-flui state is instable against perturbation an convection takes place. The value 72/C foun with the above simple reasoning on force equilibrium is totally irrelevant both careful experiments an theoretical calculations agree with Ra c = 1708 for a flui between two very large plates, the important lesson is the existence of a threshol. VIII.2.2 Toy moel for the Rayleigh Bénar instability A more refine although still crue toy moel of the transition to convection consists in consiering small perturbations ~v,, P e. aroun a static state ~v st. = ~0, st., P e.,st.,anto linearize the Boussinesq equations to first orer in these perturbations. As shown by Eq. (VIII.14), the effective pressure P e.,st. actually alreay inclues a small correction, ue to (V ) T being much smaller than 1, so that we may from the start neglect P e.. To first orer in the perturbations, Eqs. (VIII.12), projecte on the z-axis, an (VIII.13) give, after subtraction of the contributions from the static solution @v z @t = 4v z + (V ) g, @ T @t v z = 4. (VIII.17a) (VIII.17b) Moving the secon term of the latter equation to the right han sie increases the parallelism of this set of couple equations. In aition, there is also the projection of Eq. (VIII.12) along the x-axis, an the velocity fiel must obey the incompressibility conition (VIII.8). The proper approach woul now be to specify the bounary conitions, namely: the vanishing of v z at both plates impermeability conition, the vanishing of v x at both plates no-slip conition, an the ientity of the flui temperature at each plate with that of the corresponing plate; that is, all in all, 6 conitions. By manipulating the set of equations, it can be turne into a 6th-orer linear partial ifferential equation for, on which the bounary conitions can be impose. Instea of following this long roa, (54) we refrain from trying to really solve the equations, but rather make a simple ansatz, namely v z (t,~r) =v 0 e t cos(kx) which automatically fulfills the (54) The reaer may fin etails in Ref. [39, Chap. II].
126 Convective heat transfer incompressibility equation, but clearly violates the impermeability conitions, an a similar one for, with a constant. Substituting these forms in Eqs. (VIII.17) yiel the linear system v 0 = k 2 v 0 + (V ) 0 g, + k 2 v 0 g (V ) 0 =0, 0 = k 2 0 + T v T 0, v 0 + k 2 0 =0 for the amplitues v 0, 0. This amits a non-trivial solution only if + k 2 + k 2 (V ) T g =0. (VIII.18) This is a straightforwar quaratic equation for. It always has two real solutions, one of which is negative corresponing to a ampene perturbation since their sum is ( + )k 2 < 0; the other solution may change sign since their prouct k 4 (V ) T g is positive for T =0, yieling a secon negative solution, yet changes sign as T increases. The vanishing of this prouct thus signals the onset of instability. Taking for instance k = / to fix ieas, this occurs at a critical Rayleigh number Ra c = (V ) Tg 3 = 4, where the precise value (here 4 ) is irrelevant. From Eq. (VIII.18) also follows that the growth rate of the instability is given in the neighborhoo of the threshol by = Ra Ra c Ra c + k2, i.e. it is infinitely slow at Ra c. This is reminiscent of a similar behavior in the vicinity of the critical point associate with a thermoynamic phase transition. By performing a more rigorous calculation incluing non-linear effects, one can show that the velocity amplitue at a given point behaves like Ra Rac v / with = 1 (VIII.19) Ra c 2 in the vicinity of the critical value, an this preiction is borne out by experiments [40]. Such apowerlawbehaviorisagainreminiscentofthethermoynamicsofphasetransitions,more specifically here since v vanishes below Ra c an is finite above of the behavior of the orer parameter in the vicinity of a critical point. Accoringly the notation use for the exponent in relation (VIII.19) is the traitional choice for the critical exponent associate with the orer parameter of phase transitions. Eventually, a last analogy with phase transitions regars the breaking of a symmetry at the threshol for the Rayleigh Bénar instability. Below Ra c, the system is invariant uner translations parallel to the plates, while above Ra c that symmetry is spontaneously broken. Bibliography for Chapter VIII A nice introuction to the topic is to be foun in Ref. [41], which is a popular science account of part of Ref. [42]. Faber [1] Chapter 8.5 8.7 & 9.2. Guyon et al. [2] Chapter 11.2. Lanau Lifshitz [4, 5] Chapter V 49 53 & 56 57.