Chapter 7 Shear instabilities In this final Chapter, we continue our study of the stability of fluid flows by looking at another very common source of instability, shear. By definition, shear occurs whenever two adjacent fluid parcels move in parallel directions, but at different velocities. The shear is defined as the amplitude of the local velocity gradient perpendicular to the motion. Shear flows are ubiquitous in nature and can occur on any scale. Flows pumped though pipes by some pressure gradient along the pipe called Poiseuille flows) are present everywhere in natural or engineered systems: blood flow through the body, from small capillaries to arteries, fluid flow through underground river systems, magma flows and pyroclastic flows through volcano chimneys, water flowing through a hose, a kitchen faucet, oil in a car engine, in a pipeline, etc.. These are often subject to strong shear if the wall boundaries are no-slip so fluid is moving in the center of the pipe, but not on the sides. Shear flows can also be driven by differential pressure gradients or any other forces) in open systems, and are found in the ocean, in the atmospheric wind patterns, in the surface and subsurface flows of the Sun, giant planets, other stars, in the orbital motion of gas in accretion disks, etc.. The reason why shear drives instabilities is most easily understood by considering the energetics of a sheared fluid, as shown in the next Section. The technique introduced there is actually widely applicable to any fluid instabilities, but it is worth bearing in mind that it is not very rigorous nor is it meant to be), by contrast with all the other techniques introduced in the previous chapter. 7. Energetics of shear instabilities Energetic considerations for the development of instabilities are based on the following idea: given a background fluid flow, is that background the lowest possible energy state of the system, or can energy be extracted from the background by mixing material around? If it can, then instabilities are indeed energetically 39
40 CHAPTER 7. SHEAR INSTABILITIES favorable because a perturbation can use the energy extracted from the background to amplify itself, in a positive feedback loop. These ideas are illustrated below, first on unstratified shear flows, then on convection, then on stratified shear flows. 7.. Energetics of unstratified inviscid shear flows Consider a simple background fluid flow, with constant density ρ, subject to some shear. Without loss of generality, the shear can be modeled locally as ū = ūz)e x = Sze x 7.) where S is the constant) shearing rate, by moving into a suitable frame of reference. This expression can, for instance, be viewed as a Taylor-expansion of the actual shear flow near a certain point. Consider two parcels of fluid, one at z = 0, and one at z = ɛ, where ɛ is small enough for 7.) to be a good representation of the local flow. The initial total kinetic energy contained in the two parcels is E i = ρ ū0) + ρ ūɛ) = ρ S ɛ 7.) Meanwhile, the momentum of the lower parcel is zero, while that of the upper parcel is ρūɛ) = ρsɛ. Next, suppose we switch the two parcels around, but in the process, mix and equalize their momenta. If the total momentum remains conserved in the process, then the momenta of each of the two switched parcels becomes ρsɛ/, and they are both moving at the average velocity Sɛ/. The total kinetic energy of the switched parcels is then E s = 4 ρs ɛ 7.3) We see that the new background flow consisting of the switched parcels has a lower energy than the original one. The difference E = E i E s = ρs ɛ /4 > 0 is available to amplify initial perturbations and thus drive instabilities. This is the main reason why shear instabilities are so common the shear itself is a reservoir of kinetic energy that can, under some circumstances, be tapped into to drive instabilities. 7.. Energetics of stratified fluids A very similar argument can be applied to determine whether switching two elements in a stratified fluid at rest is energetically favorable or unfavorable. Indeed, consider a fluid whose density is stratified, with ρz) = ρ m + ρ 0z z 7.4)
7.. ENERGETICS OF SHEAR INSTABILITIES 4 where ρ 0z is the constant density gradient. As before, we consider two parcels, one at z = 0, and one at z = ɛ. The initial total potential energy of the two parcels is E i = 0 + ρɛ)gɛ = gɛρ m + ρ 0z ɛ) 7.5) We then switch the two parcels after equalizing their density assuming mass is conserved). The density of each parcel, after the switch, is ρ m + ρ 0z ɛ/, so the potential energy is E s = gɛ ρ m + ρ ) 0zɛ 7.6) The difference in potential energy before and after the switch is E = E i E s = gρ 0zɛ 7.7) In this case, we see that if ρ 0z < 0, then the switch is energetically defavorable: E s is higher than E i, which means that we have to pay energy to make it happen. Meanwhile, if ρ 0z > 0, then the switch is energy-favorable as in the previous section, and the remaining E can be used to drive fluid instabilities. This result is not surprising at all: in the absence of dissipation, ρ 0z < 0 corresponds to stratified fluids which are unstable to convection, as studied in the last Chapter. 7..3 Energetics of stratified shear flows Finally, we can combine these two ideas to study the energetics of stratified shear flows. Starting with the background shear 7.) and background density profile 7.4), we consider again two parcels, and exchange them after equalizing their momenta and densities respectively. The total energy before the switch is now the sum of the kinetic and potential energies of both parcels: E i = ρ m + ρ 0z ɛ)s ɛ + gɛρ m + ρ 0z ɛ) 7.8) After the switch, both parcels have density ρ m + ρ 0z ɛ/, and their momentum is the average of the momenta of the two parcels before the switch which must now be calculated more carefully since the density is no longer constant): ρ m + ρ 0zɛ ) v = ρ m + ρ 0z ɛ)sɛ 7.9) which then yields v, the velocity of both parcels after the switch: v = ρ m + ρ 0z ɛ)sɛ ρ m + ρ0zɛ The total energy after the switch is then E s = ρ m + ρ ) 0zɛ v + gɛ ρ m + ρ ) 0zɛ 4 = ρ m + ρ 0z ɛ) S ɛ ρ m + ρ0zɛ + gɛ ρ m + ρ ) 0zɛ 7.0) 7.)
4 CHAPTER 7. SHEAR INSTABILITIES The energy gained or lost is E = E i E s = g ρ 0zɛ + ρ m + ρ 0z ɛ)s ɛ 4 ρ m + ρ 0z ɛ) S ɛ ρ m + ρ0zɛ 7.) Reducing this to the same denominator, and keeping only the lowest-order terms in ɛ, we get successively E = g ρ0zɛ ρm + ρ0zɛ ) + ρ m + ρ 0z ɛ)s ɛ ρ m + ρ0zɛ ) 4 ρ m + ρ 0z ɛ) S ɛ ɛ ) ρ ms + gρ 0z ρ m + ρ0zɛ In comparison with the previous sections, this expression can be interpreted in two equivalent ways. We saw that in the absence of stratification, shear instabilities are always energetically favorable. We now see that they remain energetically favorable in the presence of an unstable density gradient ρ 0z > 0 which is really not surprising), but can be stabilized by a sufficiently strong negative background density gradient. Alternatively, 7.3) can also be interpreted to mean that instabilities on a stably stratified flow can be energetically favorable provided the shear S is sufficiently strong. 7..4 The Richardson number for stratified shear flows The results of the previous section suggest that whether stratified shear flows are stable or unstable depends on the non-dimensional ratio J = gρ 0z ρ m S = N S 7.4) recalling that N can also be written as N = gρ 0z /ρ m. This is called the Richardson number after Richardson). Because J is based on local gradients, it is often called the gradient Richardson number. Global Richardson numbers can be defined as an alternative, see later on. The Richardson number, as we saw, is related to the ratio of the potential energy lost while moving density around, to the kinetic energy gained from the shear. For instability to occur, J should be smaller than a factor of order unity, while stability is likely when J is larger than unity. It is important to realize, however, that these are just approximate arguments based on energetic considerations. They do not replace a thorough linear stability analysis, nor do they provide any rigorous results on energy stability. However, they give us a good idea of what to expect next! 7. Local linear stability analysis As for the case of convection, we now attempt to perform a local stability analysis of shear flows. We do not need to specify the origin of the background 7.3)
7.3. GLOBAL ANALYSIS OF UNSTRATIFIED PLANE PARALLEL FLOWS43 shear ū see 7.), or of the background density profile ρ see 7.4), but merely assume they exist. We also assume for simplicity that we are in the limit where the Boussinesq approximation is valid. If we let then the perturbations evolve according to ρ = ρ + ρ and u = ū + ũ 7.5) ũ = 0 + ū ũ + ũ ū = p ρge z + ν ũ t ρ m ρ = α ρ T m T t + ū T + ũ T = κ T T 7.6) where T = ρ/αρ m = T 0z e z which defines T 0z ). Once expanded into components and assuming D flow, for instance), this becomes x + w z = 0 t w w + Sz t x T t p + Sz + S w = x x + ν ũ = p z + αg T + ν w + Sz T x + wt 0z = κ T T 7.7) We immediately detect a crucial problem: this equation does not have constant coefficients, so the standard local analysis which would have us use the ansatz q = ˆq expik x x + ik z z + λt) will not work. For this reason, we press on to a global mode analysis. 7.3 Global analysis of unstratified plane parallel flows 7.3. Linear stability analysis in the inviscid limit We now consider a global model of fluid flow between two horizontal parallel plates. For simplicity, we assume in this section that there is no viscosity. As we saw in the very first lecture, this implies that any fluid flow of the form ū = ūz)e x is a solution of the momentum equation subject to the boundary
44 CHAPTER 7. SHEAR INSTABILITIES conditions w = 0 at both plates. Perturbations to this background flow satisfy x + w z = 0 t + ūz) x w w + ūz) t x + w dū dz = p x = p z 7.8) The coefficients of this system of PDEs are independent of time and of x, so we seek solutions of the form q = ˆqz) expik x x + λt). We get Eliminating ˆp, we get λ + ik x ūz)) dû dz + ik x û + dŵ dz = 0 λû + ik x ūz)û + ŵ dū dz = ik x ˆp λŵ + ik x ūz)ŵ = dˆp dz 7.9) ik x û + dŵ ) dū dz dz + ŵ d ū dz = ik x λ + ik x ūz)) ŵ 7.0) We then use the continuity equation to eliminate û, to get ) dŵ λ + ik x ūz)) dz k xŵ ik x ŵ d ū dz = 0 7.) Finally, if we define a new variable c such that λ = ik x c 7.) then the vertical velocity perturbation must satisfy ) dŵ ūz) c) dz k xŵ ŵ d ū dz = 0 7.3) This equation is called the Rayleigh stability equation. It is an ODE which needs to be solved for the function ŵz) and the eigenvalue c, subject to the boundary conditions ŵ = 0 at the top and bottom of the domain. It usually needs to be solved numerically, and is not a standard Sturm-Liouville eigenvalue problem, so that most of the theorems we have learned about the latter do not apply here. Nevertheless, there are still an interesting number of strict results that can be derived from 7.3). First, note that for any solution with k x > 0, there is another solution with k x < 0 and c c. Without loss of generality, we can therefore take k x > 0. Next, note that unstable modes for k x > 0 are characterized by c values that have a positive imaginary part. Also it s easy to show that since ŵ can be
7.3. GLOBAL ANALYSIS OF UNSTRATIFIED PLANE PARALLEL FLOWS45 complex, if ŵ is a solution then its complex conjugate ŵ is also a solution with eigenvalue c. To see this, simply take the complex conjugate of the entire equation 7.3). This means that there are only two possibilities: either the solution ŵ has c real the mode is called neutrally stable since it neither grows nor decays, but merely oscillates), or there is a complex-conjugate pair of solutions ŵ and ŵ with one of the two being an unstable mode. If c is real, then there is a possibility that the quantity ūz) c) could be zero somewhere in the domain. If this is the case, then the ODE is singular at this point. We shall return to that case later. For growing modes, however, c has a non-zero imaginary part, and the problem is regular everywhere. Based on this consideration, we can then derive one of the most important results on plane parallel shear flows: Rayleigh s inflection point theorem. This theorem states that a necessary condition for the existence of an unstable mode is that the flow profile must have an inflection point in the domain, that is, some point where ū z) = d ū/dz = 0. To show this, first assume that c has a non-zero imaginary part, and rewrite 7.3) as d ŵ dz k xŵ ŵ ūz) c = 0 7.4) ū z) Then multiply this equation by ŵ, and integrate the result over z, from the bottom boundary at z = z ) to the top boundary at z = z ). This yields z z [ ŵ d ŵ dz k x ŵ ŵ ū z) ūz) c ] dz = 0 7.5) The first term in the integral can then be integrated by parts, to give [ z ] dŵ dz + kx ŵ + ŵ ū z) dz = 0 7.6) ūz) c z The imaginary part of this equation is then Ic) z ŵ ū z) dz = 0 7.7) z ūz) c Since all of the terms within the integral are strictly positive except ū z), there are two possibilities: either ū z) changes sign somewhere in the domain, or Ic) = 0 which we had assumed is not the case). This shows that Ic) 0 requires the presence of an inflection point! Note that the presence of an inflection point, by this analysis, is only a necessary condition for instability, but not a sufficient one, which means that there could be flows with an inflection point that are still stable. However, it does imply that flows without infection point are always stable in this inviscid limit). Rayleigh s inflection point theorem has a really important, and rather peculiar consequence: it implies that linear shear flows that is, flows of the kind
46 CHAPTER 7. SHEAR INSTABILITIES described by 7.)) are linearly stable to shear instabilities! This result is quite remarkable in its stark contrast with our energetic argument of Section 7., which suggested that linear shear flows in the inviscid limit should always be unstable. As it turns out, shear flows are quite different from the previously studied case of convection in that they are subject to finite amplitude instabilities, that is, they are often linearly stable, but can be destabilized by a strong enough perturbation. More on this later. In contrast, if the flow has an inflection point, then this is usually the position around which the shear instability will develop, and where the amplitude of the perturbation is the largest. A number of additional interesting theorems can be derived from this Rayleigh s instability equation. These include: Fjortoft s theorem: This is a stronger form of Rayleigh s inflection point criterion, which states that a necessary condition for instability is that ū z)ū ū i ) < 0 somewhere in the fluid, where ū i = ūz i ) and where z i is the point at which ū z i ) = 0. Howard s semicircle theorem: This is a very useful theorem that bounds the possible values of the real and imaginary parts of c, and therefore places upper limits on the growth rate of the unstable modes. The theorem states that all unstable modes that is with Ic) > 0) have an eigenvalue c that satisfies Rc) ) max ū + min ū + Ic) max ū min ū) < 7.8) or in other words, the complex number c lies on the complex plane within the semi-circle of radius R = max ū min ū)/4, centered on the point max ū + min ū)/ on the real axis. The proof of these theorems can be found in the textbook Hydrodynamic Stability by Drazin and Reid, for instance, or in the original papers. Finally, the combination of these theorems leads to the following strong statements whose proofs are beyond the scope of this class). If ūz) is a smooth flow, then eigenmodes of Rayleigh s instability equation are of two kinds: Neutrally stable modes, with Ic) = 0. There is a continuum of them, one for each possible real) value of c in the interval [min ū, max ū]. The modes are singular at the point z where ūz) = c, and their first derivative is discontinuous there. The position of this singularity is called a critical layer. Pairs of complex-conjugate modes, with Ic) 0. There are only very few of them, at most one pair for each inflection point in the flow, and c for these complex-conjugate pairs satisfies Howard s semicircle theorem.