12.006J F2022 Lectures 17–18: Fluid Dynamics and Rayleigh-Benard Convection

Lecture notes for 12.006J/18.353J/2.050J, Nonlinear Dynamics: Chaos D. H. Rothman, MIT October 24, 2022 Contents 1 Fluid dynamics and Rayleigh-Bénard convection 1.1 The concept of a continuum . . . . . . . . . . . . 1.2 Mass conservation . . . . . . . . . . . . . . . . . . 1.3 Momentum conservation . . . . . . . . . . . . . . 1.3.1 Substantial derivative . . . . . . . . . . . . 1.3.2 Forces on fluid particle . . . . . . . . . . . 1.4 Nondimensionalization of Navier-Stokes equations 1.5 Rayleigh-Bénard convection . . . . . . . . . . . . 1.6 Rayleigh-Bénard equations . . . . . . . . . . . . . 1.6.1 Dimensional form . . . . . . . . . . . . . . 1.6.2 Dimensionless equations . . . . . . . . . . 1.6.3 Bifurcation diagram . . . . . . . . . . . . 1.6.4 Convection in the Earth . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 4 5 5 7 9 11 16 16 16 17 18 Fluid dynamics and Rayleigh-Bénard convection Reference: Tritton [1] In these lectures we derive (mostly) the equations of fluid dynamics. We then show how they may be generalized to the problem of Rayleigh-Bénard convection—the problem of a fluid heated from below. Later we show how the RB problem itself may be reduced to the famous Lorenz equations. Some topics to be discussed: • The Navier-Stokes equations • Reynolds number 1 • RB convection • Rayleigh number Thus far we have dealt almost exclusively with the temporal behavior of a few variables. In these lectures we digress, and discuss the evolution of a continous field in space and time. Aside from the central role played by research in fluid turbulence and RB convection in the development of the theory of chaos, we have another motivation: an appreciation of a hierarchy of mathematical descriptions of dynamical systems, ranging from pde’s to ode’s to discrete maps. 1.1 The concept of a continuum Real fluids are made of atoms or molecules. We could in principle write ordinary differential equations for the position and momentum of each particle. But then we’d have ∼1023 equations! The concept of a continuum allows us to write a partial-differential equation instead. We proceed to describe the essential assumption that makes this possible. Consider the following macroscopic length scales in a flow: U l1 l2 l3 In adddition to the length scales li above, we define • Lhydro : the smallest characteristic length scale of macroscopic motions. 2 • the mean free path `mfp : the characteristic length scale between molecular collisions. Fluids may be regarded as continuous fields if Lhydro  `mfp . When this condition holds, the evolution of the macroscopic field may be described by continuum mechanics, i.e., partial differential equations. To make this idea clearer, consider a thought experiment in which we measure the density of a fluid over a length scale ` using some particularly sensitive device. We then move the device in the x-direction over a distance of roughly 10`. density Suppose ` ∼ L1 ∼ `mfp . Then we expect the density to vary greatly in space as in Figure (a) below: (a) x/L1 (b) x/L 2 (c) x/Lhydro We expect that the fluctuations in (a) should decrease as ` increases. (Statistics tells us that these fluctuations should decrease like 1/N 1/2 , where N ∝ `3 is the average number of molecules in a box of size `. ) On the other hand, if ` ∼ Lhydro (see (c)), variations in density should reflect density changes due to macroscopic motions (e.g., a rising hot plume), not merely statistical fluctuations. Our assumption of a continuum implies that there is an intermediate scale, ` ∼ L2 , over which fluctuations are small. Thus the continuum hypothesis 3 implies a separation of scales between the molecular scale, L1 ∼ `mfp , and the hydrodynamic scale, Lhydro . The motion of the continuum is expressed by partial differential equations for evolution of conserved quantities. We begin with the conservation of mass. 1.2 Mass conservation Let ρ = density ~u = velocity  of a macroscopic fluid particle Consider a volume V of fluid, fixed in space: dS V u d~s is an element of the surface, |d~s| is its area, and it points in the outward normal direction. ~u is the velocity. The outward mass flux through the element d~s is ρ~u · d~s. Therefore, Z ρ~u · d~s. rate of mass loss from V = s The total mass in V is Z ρdv V Thus the rate of mass loss may be rewritten as Z Z Z ∂ρ d ρdv = − dv = + ρ~u · d~s − dt V V ∂t s 4 Shrinking the volume, we eliminate the volume integrals and obtain Z  ∂ρ = − lim ρ~u · d~s/V . V →0 ∂t Recall that the RHS above is the definition of the divergence operator. We thus obtain ∂ρ ~ · (ρ~u) = −∇ ∂t We see that to conserve mass, a net divergence creates a corresponding change in density. For incompressible fluids, ρ ∼ constant. (This result is not an assumption, but instead derives from the assumption that the Mach number, the square of the ratio of the fluid velocity to the speed of sound, is much less than unity.) Then ~ · ~u = 0. ∇ which is the equation of continuity for incompressible fluids. 1.3 Momentum conservation We seek an expression of Newton’s second law: d (momentum of fluid particle) = force acting on fluid particle dt 1.3.1 (1) Substantial derivative We first focus on the LHS of (1). There is a conceptual problem: d (particle momentum) cannot be given at a dt fixed location, because • the momentum field itself changes with respect to time; and 5 • fluid particle can change its momentum by flowing to a place where the velocity is different. To better understand this problem physically, consider how a scalar property— the temperature T —of a fluid particle changes in time. A small change δT is produced by small changes δt in time and δx, δy, δz in the position of the fluid particle: δT = ∂T ∂T ∂T ∂T δt + δx + δy + δz ∂t ∂x ∂y ∂z Divide by δt to obtain the rate of change: δT ∂T ∂T δx ∂T δy ∂T δz = + + + δt ∂t ∂x δt ∂y δt ∂z δt In the limit δt → 0, δx → ux , δt δy → uy , δt δz → uz δt The rate of change of T of a fluid particle is then ∂T ∂T ∂T DT ∂T = + ux + uy + uz Dt ∂t ∂x ∂y ∂z = ∂T ~ + ~u · ∇T ∂t where ∂ D ~ = + ~u · ∇ Dt ∂t is the substantial derivative or convective derivative operator. Thus we see that the temperature of a fluid particle can change because • the temperature field changes “in place” (via ∂/∂t); and • the particle can flow to a position where the temperature is different (via ~ ~u · ∇). 6 Note that the same analysis applies to vector fields such as the velocity ~u: D~u ∂~u ~ u = + (~u · ∇)~ Dt ∂t Therefore the velocity ~u enters D~u/Dt in 2 ways: • ~u changes (in place) as the fluid moves (∂/∂t) ~ • ~u governs how fast that change occurs (~u · ∇). This dual role of velocity is the essential nonlinearity of fluid dynamics and thus the cause of turbulent instabilities. We can now express the rate-of-change of momentum per unit volume (i.e., LHS of (1)): ∂~u D~u ~ u =ρ + ρ(~u · ∇)~ ρ Dt ∂t ρ is outside the differential because a fluid particle does not lose mass. Density changes thus mean volume changes, which are irrelevant to the momentum change of that particle. Above we have written the (rate of change of momentum) per unit volume, which need not be equal to the rate of change of (momentum per unit volume). 1.3.2 Forces on fluid particle To obtain the full dynamical equation, we need the RHS of ρ D~u = Force acting on fluid particle / unit volume. Dt These forces are • body force (i.e., gravity) • pressure • viscous friction (internal stresses) 7 Body force: We represent the externally imposed body force (per unit volume) by F~ . Pressure: Fluid flows from high to low pressure. Thus ∂p pressure force = − unit volume ∂x in 1-D ~ = −∇p in 3-D Viscous friction: Viscous stresses are the source of dissipation in fluids. They resist relative movements between fluid particles. For example, the shear flow y u u x is resisted more by high viscosity fluids than low viscosity fluids. This resistance derives from molecular motions. (A nice analog is Reif’s picture of two mail trains, one initially fast and the other initially slow, that trade mailbags.) In the simple shear flow above, random atomistic motions result in a flux of x-momentum in the y-direction. In Newtonian fluids, this flux, which we call Pxy , is proportional to the velocity gradient: ∂ux Pxy = −η ∂y where η is called the dynamic viscosity. η has units of mass/(length × time). The shear stress can occur at any orientation. Analogous to the 1-D Newto8 nian condition above, we define the viscous momentum flux Pij = −η ∂ui . ∂xj The conservation of momentum requires that the divergence of the momentum flux Pij be balanced by a change in the momentum of a fluid particle. Loosely stated, ∂(ρui ) ∂t ~ · Pij = − = −∇ viscous We thus find that X ∂ X ∂2 Pij = η 2 ui ∂x ∂x j j j j viscous force = η∇2~u. unit volume (A careful derivation requires consideration of the tensorial relationship between viscous stress and the rate of deformation.) Newton’s second law then gives the Navier-Stokes equation for incompressible fluids: ∂~u ~ u ρ + ρ(~u · ∇)~ ∂t | {z } ~ + η∇2~u −∇p | {z } = (mass per unit vol)×acceleration + stresses on fluid element per unit vol F~ |{z} body force per unit vol Incompressibility arose from our negelect of compressive forces on fluid elements. 1.4 Nondimensionalization of Navier-Stokes equations Define the characteristic length scale L and velocity scale U . We obtain the non-dimensional quantities x0 = ~u0 = x , L ~u , U y , L z0 = z L U t0 = t , L p0 = p ρU 2 y0 = 9 The dynamical equations (without body force) become ~ 0 · ~u0 = 0 ∇ ∂~u0 0 ~0 0 ~ 0 p0 + 1 ∇02~u0 + (~ u · ∇ )~ u = − ∇ ∂t0 Re where Re = Reynolds number = ρU L η is the dimensionless control parameter. The Reynolds number quantifies the relative importance of the nonlinear term to the viscous term. To see why, note the following dimensional quantities: ~ u| ∼ |ρ~u · ∇~ |η∇2~u| ∼ Their ratio is ρU 2 L ηU L2 nonlinearity dissipation ~ u| |ρ~u · ∇~ ρU L ∼ = Reynolds number |η∇2~u| η High Re is associated with turbulence (i.e., nonlinearities). Low Re is associated with laminar or creeping flows dominated by viscous friction. Note that as long as Re remains the same, the dimensional parameters like U and L can change but the the flow (i.e., the equation it solves) does not. This is dynamical similarity. An example is running vs. swimming:   η = 0.15 cm2 /sec and ρ air   η ρ = 0.01 cm2 /sec water On the other hand, comparing 100 meter world records, Urun Uswim 104 cm ∼ = 103 cm/sec 10 sec 104 cm ∼ ∼ 2 × 102 cm/sec 50 sec 10 Taking L ∼ 100 cm, Re(swim) ∼ 2 × 104 and Re(run) ∼ 6 × 103 Thus for both swimming and running, Re ∼ 104 , well into the turbulent regime. Surprisingly, despite the slower speed of swimming, Re(swim) is somewhat greater. Another example: bacteria swimming in water is roughly like us swimming in molasses, since the the small size and slow speed of bacteria would correspond to a larger and faster body in a more viscous fluid. 1.5 Rayleigh-Bénard convection In a thermally expansive fluid, hot fluid rises. R-B convection concerns the study of the instabilities caused by rising hot fluid and falling cold fluid. Typically, fluid is confined between two horizontal, heat-conducting plates: T0 T=T 0 g d (cold) pure conduction fluid T=T 0 + δ T T=T 0 + δ T (hot) temperature In the absence of convection—the transport of hot fluid up and cold fluid down—the temperature gradient is constant. Two cases of interest: • δT small: no convective motion, due to stabilizing effects of viscous friction. 11 • δT large: convective motion occurs. How large is a “large δT ” ? We seek a non-dimensional formulation. The following fluid properties are important: • viscosity • density • thermal expansivity • thermal diffusivity (heat conductivity) Convection is also determined by • d, the box size • δT (of course) Consider a small displacement of a cold blob downwards and a hot blob upwards: T=T 0 T=T 0 + δ T Left undisturbed, buoyancy forces would allow the hot blob to continue rising and cold blob to continue falling. There are however damping (dissipation) mechanisms: • diffusion of heat 12 • viscous friction Let DT = thermal diffusivity, which has units length2 [DT ] = time The temperature difference between the two blobs can therefore be maintained at a characteristic time scale d2 τth ∼ DT We also seek a characteristic time scale for buoyant displacement over the length scale d. Let ρ0 = mean density ∆ρ = −αρ0 ∆T, α = expansion coefficient Setting ∆T = δT , buoyancy force density = |~g ∆ρ| = gαρ0 δT. Note units: [gαρ0 δT ] = mass (length)2 (time)2 The buoyancy force is resisted by viscous friction between the two blobs separated by ∼ d. The viscous friction between the two blobs diminishes like 1/d (since viscous stresses ∝ velocity gradients). The rescaled viscosity has units hη i mass = d (length)2 (time) Dividing the rescaled viscosity by the buoyancy force, we obtain the characteristic time τm for convective motion: τm ∼ η/d η = . buoyancy force gαρ0 d δT 13 Convection (sustained motion) occurs if time for motion < diffusion time for temperature difference τm < τth Thus convection requires τth > constant τm or ρ0 gαd3 δT ≡ Ra > constant ηDT Ra is the Rayleigh number. A detailed stability calculation reveals that the critical constant is 1708. Our derivation of the Rayleigh number shows that the convective instability is favored by • large δT , α, d, ρ0 . • small η, DT . In other words, convection occurs when the buoyancy force ρ0 gαd3 δT exceeds the dissipative effects of viscous drag and heat diffusion. Note that box height enters Ra as d3 . This means that small increases in box size can have a dramatic effect on Ra. For Ra sufficiently large, the flow becomes turbulent. Some examples (from Prof. Jun Zhang, NYU): Here the gray scale is related to the thermal gradient. Image courtesy of Prof. Jun Zhang, NYU. Used with permission. 14 Here the viscous flow moves the floating boundary and the the boundary affects the flow, an interplay roughly analogous to fluid motions beneath tectonic plates. Image courtesy of Prof. Jun Zhang, NYU. Used with permission. A close-up (red is cool, blue is warm). Image courtesy of Prof. Jun Zhang, NYU. Used with permission. And here’s a picture of downgoing cold plumes (red) plunging from the upper thermal boundary layer into the warm (blue) fluid below: Zocchi et al. [2] Courtesy Elsevier, Inc., http://www.sciencedirect.com. Used with permission. 15 1.6 1.6.1 Rayleigh-Bénard equations Dimensional form We employ the Boussinesq approximation: density perturbations affect only the gravitational force. The momentum equation is therefore the Navier-Stokes equation augmented by the buoyancy force: ∂~u ~ u = − 1 ∇p ~ + ν∇2~u − ~g α(T − T0 ) + ~u · ∇~ ∂t ρ0 Here we have written the kinematic viscosity ν = η/ρ0 The mass conservation equation is again ~ · ~u = 0. ∇ We now additionally require an equation for the convection and diffusion of heat: ∂T + (~u · ∇)T = DT ∇2 T. ∂t 1.6.2 Dimensionless equations The equations are nondimensionalized using length scale = d time scale = d2 /DT temperature scale = δT /Ra. An additional dimensionless parameter arises: Pr = Prandtl number = ν/DT , which is essentially the ratio of momentum diffusion to thermal diffusion. 16 We also use the dimensionless temperature fluctuation θ = deviation of dimensionless T from the simple conductive gradient Dropping primes, the mass conservation equation is ~ · ~u = 0. ∇ Momentum conservation yields (ẑ is the unit vector pointing up)   1 ∂~u ~ u = −∇p ~ + θẑ + ∇2~u. + ~u · ∇~ Pr ∂t The heat equation becomes ∂θ ~ = Ra(~u · ẑ) + ∇2 θ + ~u · ∇θ ∂t Note that there are two nonlinear terms: ~u • ~u · ∇~ ~ • ~u · ∇θ Their relative importance depends on Pr: ~ u dominates. Instabilities are “hydrodynamic.” • small Pr ⇒ ~u · ∇~ ~ dominates. Instabilities are thermally induced. • large Pr ⇒ ~u · ∇θ 1.6.3 Bifurcation diagram For Ra < Rac , there is no convection. For Ra > Rac , but not too large, a regular structure of convection “rolls” forms, with hot fluid rising and cold fluid falling: 17 T = T0 d T = T0 + δ d Now imagine placing a probe that measures the vertical component v of velocity, somewhere in the box midway between the top and bottom. A plot of v(Ra) looks like v+ v convection (stable) 0 conduction (unstable) conduction rest Ra c Ra v− The transition from conduction to convection is therefore a supercritical pitchfork bifurcation. Note that at any particular location we cannot know in advance whether the symmetry is broken by an upgoing or downgoing velocity. 1.6.4 Convection in the Earth The Earth’s radius is about 6378 km. It is layered, with the main divisions being the inner core, outer core, mantle, and crust. The Earth’s crust—the outermost layer—is about 30 km thick. The mantle ranges from about 30–2900 km. The mantle is widely thought to be in a state of thermal convection. The 18 source of heat is thought to be the radioactive decay of isotopes of uranium, thorium, and potassium. Another heat source is related to the heat deriving from the gravitational energy dissipated by the formation of the Earth roughly 4.5 Ga. At long time scales mantle rock is thought to flow like a fluid. However its effective viscosity is the subject of much debate. One might naively think that the huge viscosity would make the Rayleigh number quite small. Recall, however, that Ra scales like d3 , where d is the “box size”. For the mantle, d is nearly 3000 km!!! Consequently Ra is probably quite high. Current estimates suggest that 3 × 106 . Ramantle . 109 which corresponds to roughly 103 × Rac . Ramantle . 106 Rac The uncertainty derives principally from the viscosity, and its presumed variation by a factor of about 300 with depth. Regardless of the uncertainty, we can conclude that Ra for the mantle is more than sufficient for convection, and therefore that convection is likely the driving force of plate tectonics and volcanism. It turns out that volcanism is, over the long-term, responsible for the CO2 in the atmosphere, and thus the source of carbon that is fixed by plants. (Weathering reactions remove C from the atmosphere.) Thus in some sense thermal convection may be said to also sustain life. That is, without convection, there probably would be no CO2 in the atmosphere, and therefore we wouldn’t be around to discuss it... 19 References 1. Tritton, D. J. Physical Fluid Dynamics, 2nd edition (Clarendon Press, Oxford, 1988). 2. Zocchi, G., Moses, E. & Libchaber, A. Coherent structures in turbulent convection, an experimental study. Physica A: Statistical Mechanics and its Applications 166, 387–407 (1990). 20