Selected Topics in Fluid Dynamics & Advanced Hydrodynamics Flows in rotating systems (Dated: May 9, 2020)
Recommended reading: Lautrup (Chapter 20) & Acheson (Chapter 8.5)
The motivation for studying rotating flows is threefold. Firstly, it is important to assess the effect of rotation on flows in an industrial context. Secondly, and more importantly, we often witness flows in rotating frames of reference in our everyday experience. The effects of rotation are particularly pronounced in geophysical flows. As observers of these, we are necessary in a non-inertial frame of reference Even if the corrections to acceleration for such an observer are small compared to gravity, for large scale, slow flows, such as in the ocean or atmosphere, they can shape the flows we see in a considerable way. Thirdly, astrophysical flows are known to be deeply impacted by the effects of (often fast) rotation of their components. The main new players are two ficticious forces (they are ficticious from the point of view of an intertial observer, but embark on a carousel to convince yourself that they can be painfully real!): the centrifugal force and the Coriolis force. Thus, when in a non-inertial, rotating frame, we need to correct the Newton’s second law by incorporating these two forces. Here, we will only consider steady rotation, so Ω = const. For unsteady rotation, revisiting any Classical Mechanics lecture will be helpful to include dΩ/dt terms.
VELOCITY AND ACCELERATION IN STEADY ROTATION
In Exercises, we will discuss the derivation of the equations of motion in a steadily rotating frame of reference. Considering two frames, one inertial (x0, y0, z0) and one rotating (x, y, z). By examining the acceleration in the rotating frame, we find that
a = a0 − 2Ω × u − Ω × (Ω × u) (1)
Question: Estimate the magnitude of these terms (as compared to gravity as the natural scale for acceleration) for a carousel with R = 5 m and rotating at one turn per 6 seconds. Are they significant?
We will now asses their significance for Earth. At typical values the centrifugal force can be estimated as Ω2R, with R being now the Earth’s radius, which compares to gravity as Ω2R/g ≈ 1/291, so it is negligibly small. It leads to a slight flattening of the rotating planet 2 perpendicularly to its rotation axis, but in the geophysical context it does not play a major role and, as we will show, it can be incorporated into effective pressure. The Coriolis force is different. Let us consider a point on the surface of the Earth at a latitude θ and introduce a local flat-Earth coordinate system with the x-axis pointing towards the East, y pointing North, and z being vertical. The rotational velocity of the Earth then is Ω = Ω(0, sin θ, cos θ). The Coriolis acceleration gC in terms of the local velocity u = dx/dt reads uy cos θ − uz sin θ C g = 2Ω −ux cos θ . (2) ux sin θ At the poles the Coriolis force is always horizontal, and at the equator it is always vertical for purely horizontal motion. The vertical component of the Coriolis force is very small compared to gravity. For example, for a jet aircraft flying westwards at middle latitudes at a speed close to the velocity of sound, it amounts to about 0.3% of gravity, similarly to the centrifugal force. Thus it can be neglected for most Earthly flows of interest. For typically small vertical flow velocities, we will also neglest the Coriolis force due to uz. So we will adopt an assumption that uy C g = 2Ω⊥ −ux , (3) 0 with Ω⊥ = Ω cos θ. Concluding, in a flat-Earth coordinate system the Coriolis force indeed looks as if the Earth was rotating about the local vertical with the angular velocity Ω⊥. At middle ◦ − −1 latitudes θ ≈ 45 , we can take Ω⊥ ≈ 10 4 s , rendering the typical period of rotation of a
Foucault’s pedulum [1] being 2π/Ω⊥ ≈ 34 h.
FLOW IN ROTATING FRAMES
Applying the modified acceleration to the Navier-Stokes equation (exercise), we can show that the resulting governing equations for the flow become ∂u 1 + u · ∇u + 2Ω × u − Ω × (Ω × u) = − ∇p + ν∇2u, (4) ∂t ρ ∇ · u = 0. (5)
Moreover, upon defining the effective pressure P (another exercise), the flow equations may be recast as ∂u 1 + u · ∇u = −2Ω × u − ∇P + ν∇2u, (6) ∂t ρ ∇ · u = 0, (7) 3 where for the calculation of the Coriolis term we can only use the local, vertical angular velocity
Ω⊥.
Dimensionless numbers: Reynolds, Rossby, and Ekman
It is again interesting to study the limits of this equation. Taking the flow velocity scale U and the length scale L, we can define, as usually, the Reynolds number
UL Re = . (8) ν
In the limit of large Reynolds number, Re 1, the advective term dominates over the viscous term, and we can generally neglect the effects of viscosity and the last term in Eq. (6). Remember though, that this approximation will generally only work far away from boundaries, where boundary layers will be formed! From the remaining terms, it is tempting to compare the relative importance of the advective term and the Coriolis acceleration, which leads to (exercise) the Rossby number
|u · ∇u| U Ro = ≈ , (9) 2|Ω × u| 2ΩL named after a Swedish-born US meteorologist Carl-Gustav Arild Rossby (1898-1957). In this idealised case, the Coriolis force will only be significant for low Ro, so for U . 2ΩL, when the translational flow velocity is comparable or smaller to the local rotation-induced velocity. Example Consider ocean currents and weather cyclones, which are relatively steady rotating flows. With U ∼ 1 m/s (ocean) or U ∼ 10 m/s (wind), and the typical local angular velocity of 2Ω ≈ 10−4 Hz, we get Rossby numbers of Ro ≈ 0.01 (ocean) or Ro ≈ 0.1 (cyclones). We conclude that both are thus dominated by the Coriolis force, which plays the key role in their evolution. For a typical swimming human, with L ∼ 1 m and U ∼ 1 m/s, we find Ro ≈ 104, which justifies the complete neglect of the Coriolis force in this case. Interestingly, in systems where gravity is simulated by rotation (such as a space station), the Coriolis force will cause surprises when playing ball games (see Lautrup). For lower Reynolds numbers, it becomes important to compare the relative strength of viscous and Coriolis effects. The relevant dimensionless combination (exercise) is the Ekman number
|ν∇2u| ν Ro Ek = = = , (10) 2|Ω × u| 2ΩL2 Re named after a Swedish oceanographer Vagn Walfrid Ekman (1874– 1954). When the Ekman number is small, the viscous force is small compared to the Coriolis force. This is automatically the case for moderate Ro and large Re. In the sea and the atmosphere, the huge Reynolds number 4 makes the Ekman number negligible but, warned by our experience with boundary layers, we may expect the Ekman number to be larger close to boundaries. Indeed, it normally is of order unity close to bounding surfaces where viscosity dominates over advection, and competes with the Coriolis force. This gives rise to a particular boundary layer, called the Ekman layer, which we will discuss further.
GEOSTROPHIC FLOW
In most natural flow context, we may neglect both the Rossby ansd Ekman numbers, since Ro 1 and Ek 1. This means we are disregarding both viscous and advective terms in the Navier-Stokes equations. The flow is completely dominated by the Coriolis force and called geostrophic. For a steady flow, we end up with a particularly simple equation 1 − 2Ω × u − ∇P = 0, (11) ρ which has a nice hydrostatic analogy – the effective pressure gradient must always balance the Coriolis force. However, the Coriolis force has a rather special form, and this has far-reaching consequences. The most spectacular one is that geostrophic flow is essentially two-dimensional. To examine this, notice that
u · ∇P = 0, (12) which implies that the effective pressure is constant on streamlines. Moreover, if the motion is horizontal, the gravitational force plays no role in the pressure, and effective pressure is the same as the hydrostatic pressure. This means that streamlines and isobars coincide in geostrophic flow. Thus, in a typical weather map, the direction of the wind can be readily read off from the isobars. Now, to account for the sign of the wind direction, we may use our knowledge of the Coriolis force direction, which makes winds in the Northern hemisphere turn anti-clockwise around the low pressure areas (cyclones) and clockwise around the high pressure areas (anti-cyclones). The wind velocity may be determined from the geostrophic equation (11) using Ω · u = 0 to find Ω × ∇P u = . (13) 2Ω2ρ In the absence of the Coriolis force, the air flow would happen along the pressure gradients only, from the high to low pressure regions. The 2D character of geostrophic flow may be seen as follows. Taking a scalar product of Eq. 11 with Ω, we find ∂P (Ω · ∇)P = Ω = 0, (14) ∂z 5 so the effective pressure is constant along the axis of rotation. In a constant gravitational field, using the definition of P and again disregarding the centrifugal contribution, we find the hydrostatic pressure as
p = P (x, y) − ρgz. (15)
The fact that the effective pressure is invariant under translations along the rotation axis extends to the whole flow. Taking the curl of Eq. (11) and using incompressibility, we find
∂u 0 = ∇ × (Ω × u) = Ω(∇ · u) − (Ω · ∇)u = −Ω . (16) ∂z
Thus the flow field u is a function of x and y only and is two-dimensional! This result is called the Taylor-Proudman theorem. To the extent that weather cyclones satisfy the conditions for steady geostrophic flow, we conclude that the same large-scale wind patterns are found all way up through the atmosphere. Taylor columns The Taylor-Proudman theorem strangely states that if one disturbs a rotating fluid at say, z = 0, then the disturbance pattern will be copied to all values of z (after a transient period of time evolution). So even if the disturbing object has a finite extent in z, a so-called Taylor column (of disturbed flow) is created in the rotating fluid. Taylor columns are sometimes also called Proudman pillars. Taylor first confirmed this experimentally in 1923. Though the mechanics of the formation of Taylor columns remain complicated, this is an important result e.g. for studies of ocean circulation. Check out the Videos: Amazing lab demonstration from UCLA https://www.youtube.com/watch?v=7GGfsW7gOLI Taylor columns in the Southern Oceans https://www.youtube.com/watch?v=SOnm58ajwjY
THE EKMAN LAYER
In a nearly ideal fluid flow around a body, boundary layers must arise, since viscosity has to step in to satisfy the no-slip boundary condition on the surface. Typically, boundary layers are thinnest at the leading edge of the body and extend further towards the rear [cf. the fore-aft flow symmetry in Stokes flows]. The growth of these layers is a separate question, but for now we will focus on the particular case of the Ekman boundary layer in a rotating system. If the said body is immersed in a geostrophic flow, boundary layers must also arise. For small bodies, the effect of the rotation will be negligible, but when the Rossby number becomes of order unity, the Coriolis force starts playing a role in the formation of boundary layers. As the flow has to 6 rise from zero on the surface to its matching geostrophic-flow value, the Coriolis force becomes more pronounced, turning the fluid by moving it to the right (for anti-clockwise rotation). This geostrophic cross-wind confines the boundary layer, called the Ekman layer, to a finite thickness. To analyse the Ekman layer, let us assume that there is a steady geostrophic flow in the x- direction, with a velocity ux = U and an effective pressure P = −2ρΩUy, with a gradient in the y direction (a flow similar to the example of canal flow from Lautrup, p. 283). The flow is above a steady no-slip boundary at z = 0. We will exploit the symmetry of the system to find a solution interpolating between a stationary boundary and the geostrophic flow. Ignoring centrifugal acceleration, we see that the equations of motion are independent of the exact position (x, y) and thus invariant under translations in this plane. We then guess that a solution which would have this property could be u = u(z), which depends only on the height z. Under this assumption, mass conservation implies ∂zuz = 0, which means constant vertcial velocity.
Since at z = 0 the no-slip condition requires vz = 0, then it vanishes everywhere. The flow in the transition layer between the geostrophic flow and the no-slip surface is thus horizontal, independent of x and y, and varies with z. The advective term vanishes as
u · ∇u = (ux∂x + uy∂y)u(z) = 0, (17) and the Navier-Stokes equations (with the Coriolis force) become
1 ∂P d2u 0 = 2Ωu − + ν x , (18) y ρ ∂x dz2 1 ∂P d2u 0 = −2Ωu − + ν y , (19) z ρ ∂y dz2 1 ∂P 0 = − . (20) ρ ∂z
From the last equation, we see that P is independent of height z and for all heights it is equal to its geostrophic value P = −2ρΩUy outside the boundary layer. With this result, the equations simplify to
d2u ν x = −2Ωu , (21) dz2 y d2u ν x = −2Ω(U − u ). (22) dz2 x
This is a pair of homogeneous differential equations for U − ux and uy. From the first, we 2 2 get uy = −(ν/2Ω)d ux/dz , which inserted into the second equation gives a single fourth-order equation
d4(U − u ) 4Ω2 x = − (U − u ). (23) dz4 ν2 x 7
The general solution of this linear fourth-order differential equation will be a linear combination of four terms of the form ekz, where k4 = −4Ω2/ν2. Defining now
r ν δ = , (24) Ω the roots of k4 = −4/δ4 will be
1 ± i k = ± . (25) δ
Of these roots, those with a positive real part can be disregarded, since these would grow exponentially as ekz with z → ∞. So the most general solution will be
−(1+i)z/δ −(1−i)z/δ U − ux = Ae + Be , (26)
2 2 and, since uy = −(ν/2Ω)d ux/dz
−(1+i)z/δ −(1−i)z/δ uy = i Ae − Be . (27)
In this solution, A and B are unknown integration constants, which can be determined from the no-slip boundary condition ux = uy = 0 at z = 0 and read A = B = U/2. Finally, we find
−z/δ ux = U 1 − e cos z/δ , (28)
−z/δ uy = Ue sin z/δ. (29)
As we see from the above, δ is a measure of the thickness of the Ekman layer. It is independent of U, so it has the same thickness even if the geostrophic flow exhibits a spatial variety of its speed. In Figure1 (from Lautrup) the velocity components are plotted as a function of scaled height z/δ in units of the asymptotic flow U. One notes that ux first overshoots its asymptotic value, and then quickly returns to it. The y-component also oscillates but is π/2 out of phase with the x-component. The direction of the velocity close to the wall at z = 0 is at an angle π/4 to the left of the asymptotic geostrophic flow. Plotted parametrically as a function of height, the velocity components create a characteristic spiral, called the Ekman spiral, shown on the right. The damping is, however, so strong that only the very first turn in this spiral is visible. The presence of an Ekman layer of the right thickness has also been confirmed by laboratory experiments. For the atmosphere at middle latitudes the thickness becomes δ = 55 cm when the diffusive viscosity ν = 1.54·10−5 m2s−1 is used. This disagrees with the measured thickness of the Ekman layer in the atmosphere which is more like a kilometre. The reason is that atmospheric flow tends to be turbulent rather than laminar with an effective viscosity that can be up to a million times larger than the diffusive viscosity [2]. But this is a whole another story. 8
Figure 1. The Ekman velocity profile, taken from Lautrup’s book.
[1] Wiki: https://en.wikipedia.org/wiki/Foucault_pendulum. [2] J. Pedlosky: Geophysical Fluid Dynamics, Springer, 1987.