When one contemplates the one-parameter family of steady inviscid shear flows discovered by J. T. Stuart in 1967, an obvious thought is that these flows resemble a row of vortices diffusing in a viscous fluid, with the parameter playing the role of a reversed time. In this paper, we ask how close this resemblance is. Accordingly, the paper begins to explore Navier–Stokes solutions having as initial condition the classical, irrotational flow due to a row of point vortices. However, since we seek explicit answers, such exploration seems possible only in two relatively easy cases: that of small time and arbitrary Reynolds number and that of small Reynolds number and arbitrary time.
The classical, steady, irrotational flow due to a row of point vortices has stream function (Lamb 1932, p. 224)(1.1)The cat's-eye pattern implied by this function and made famous by Kelvin (Thomson 1880) is shown in figure 1. The vortices are at the points (2nc, 0), n∈, where c is a positive constant; each has clockwise circulation κ>0, so that the fluid velocityStuart (1967) made this classical flow an endpoint of a one-parameter family of steady inviscid (rotational) shear flows. The stream functions are(1.2)whence(1.3a)(1.3b)The equations of steady motion are satisfied because the vorticity on each streamline is constant (while varying from one streamline to the next when α<1),(1.4)where Δ denotes the Laplace operator. As α decreases from 1, the cat's eyes become flattened (figure 2) and the velocity becomes less dependent on x; at α=0, there results the parallel shear flow with velocity(1.5)Figure 3 shows profiles of horizontal velocity, for various values of α, on the vertical line through a point vortex or centre of vorticity. We note that, as y→0, u(0, y; α)→±∞ like 1/y if α=1, but →0 like y if 0≤α<1.
We define a period strip S by S≔(−c,c]×, where denotes the real line, and observe that the total vorticity in S is independent of α,(1.6)as one sees by considering the circulation around the boundary of a large rectangle (−c, c)×(−M, M) and letting M→∞.
These observations make it natural to visualize the Stuart family of flows as a row of vortices diffusing in a viscous fluid, with α playing the role of a reversed time. In other words, one might expect that, if the classical vortex row corresponding to α=1 provides the initial condition for a viscous flow, then that flow should evolve with increasing time rather as the Stuart family of flows changes with decreasing α. This notion has been used by Newton & Meiburg (1991), who postulate a precise function , where denotes a dimensionless time, and then interpret (1.2) as the evolving stream function of a viscous flow. On the other hand, the vorticity of the Stuart flows is an even function both of x and of y, for all values of α, and this is unlikely to be the case for a true row of vortices evolving in a viscous fluid.
In this paper we ask: what is the nature of Navier–Stokes solutions having the classical flow (1.1) as initial condition? In particular, how good an approximation to such solutions is the (much simpler) Stuart family of flows under a suitable choice of ? Of course, for arbitrary Reynolds number κ/ν (where ν denotes the kinematic viscosity of the fluid) and for arbitrary time this question is much too hard, since we are demanding an explicit answer. However, for arbitrary Reynolds number and small time, or for small Reynolds number and arbitrary time, something explicit can be done. In this paper, we assemble some of the ingredients and machinery for these small versions of the Navier–Stokes problem, then we take preliminary steps towards their solution. A full treatment of even these small problems will require much greater length.
2. The Navier–Stokes problem
Notation and terminology. Points of the plane 2 will be denoted by z≔(x, y), where convenient. We shall use t≔νT, where ν is the kinematic viscosity and T denotes the physical time; this simplifies the heat operator in (2.2) and many subsequent formulae. The word periodic will mean: having period 2c as x varies; thus f: 2→ is periodic if (and only if) f(x+2c, y)=f(x, y) for all (x, y)∈2. The periodic Green function of the Laplace operator Δ is(2.1)it satisfieswhere δ denotes the Dirac generalized function. We have already encountered G(z) in (1.1). Recall that S≔(−c,c]×.
Our Navier–Stokes problem is to find a periodic vorticity ω(z, t) and a stream function ψ(z, t) such that(2.2)(2.3)(2.4)Note that (2.1) and (2.3) make ψ periodic. For a periodic solution ω of (2.2)–(2.4), the total vorticity in S is conserved: (1.6) holds for all t≥0, as one shows by integrating (2.2) over S and using periodicity.
As was noted in §1, at present we can hope for explicit results only in the following two cases: (i) arbitrary Reynolds number κ/ν and small t (where either small ν or small physical time T makes t small) and (ii) small Reynolds number κ/ν and arbitrary t. In both the cases, the nonlinear right-hand member of the vorticity equation (2.2) can be neglected at the first step (is negligible to the lowest order). In case (i), the dominant parts of the vorticity and velocity, as t↓0, are those of an isolated diffusing (circular) vortex, for which u.∇ω vanishes identically (because, when the vortex is centred at the origin, the velocity (ψy, −ψx) is proportional to (y, −x), while the vorticity gradient (ωx, ωy) is proportional to (x, y).) In case (ii), if we were to use non-dimensional variables, then the right-hand member of (2.2) would have a factor κ/ν relative to the left-hand member.
3. The problem with small time: first steps
We expect that, in the period strip S and with r≔(x2+y2)1/2,(3.1)and this is exponentially small unless r2 is O(t). Therefore, we introduce inner variables(3.2)(where ∂/∂τ means that ρ and θ are held constant), and pose(3.3)where . The first three terms provide inner approximations (τ↓0 with ρ fixed, so that r↓0), but ω* must be shown to exist on for some and to be suitably small. The equation governing is the linearization of (2.2) about and about the velocity induced by . The forcing term is due to the dominant part, (κπ/24c2)(y, x), of the irrotational velocity induced by the other vortices (by the vortices at (2nc, 0), n≠0) and the gradient of . This equation is(3.4)where we have used and where(3.5a)(3.5b)(3.5c)with ζ≔(ρ cos θ, ρ sin θ). A transformation and an a priori bound for q lead to the following result.
Equation (3.4) has a smooth, bounded, unique and semi-explicit solution andA similar result holds for . The problem for the remainder ω* is considerably more difficult, but there are grounds for optimism, because in an analogous problem (Fraenkel & McLeod 2003) we have an inequality (an a priori bound for the analogue of ω*) that we believe to be the essential step.
4. Small Reynolds number: the Stokes approximation
Neglecting the right-hand member of (2.2) and labelling this approximation with subscript 0, we can represent the vorticity as a row of heat sources,(4.1)where . We have already encountered the term with n=0 in (3.1). The stream function and velocity corresponding to (4.1) are easily found from consideration of a single diffusing vortex; it is worthwhile to sum the series for the irrotational part of the velocity field by reference to (1.1). Thus(4.2)where(4.3)(4.4a)(4.4b)These heat-source series converge rapidly when t is not large, say 0<t≤c2; they will be put to work in §5. We note in passing that(4.5)where ϑ3 is a theta function in the notation of Whittaker & Watson (1927, pp. 464 and 475); also that 2I(ρ)=Ei(ρ2) in the notation of Whittaker & Watson (1927, p. 352) for the exponential integral (which enjoys many different notations). Neither of these facts is of great importance here. On the other hand, it will be useful that the irrotational part, say irrot(u0, v0), of the velocity field (4.4a) and (4.4b) can be written (Lamb 1932, p. 224) as(4.6)where z=x+iy and λ≔π/(2c). (As is common, we identify z=(x, y)∈2 and z=x+iy∈.)
For large time, say c2≤t<∞, Fourier series are more useful than those in (4.1)–(4.3), (4.4a) and (4.4b). One finds that(4.7)either by using a Laplace transformation with respect to t in order to solve the heat equation with initial condition (2.4), or by use of (4.5). Withit follows that(4.8a)where(4.8b)and, for n=1, 2, 3, …,(4.8c)The term κh0(t) in (4.8a) is included only to make (4.2) and (4.8a) describe the same function; h0 can be calculated, but it is of little importance because it does not contribute to the velocity. From (4.8a)–(4.8c), or more simply from (4.7), one finds that(4.9)For a comparison with the ultimate velocity (1.5) of the Stuart family of flows, we cheat: in order that uy at y=0 be the same for (1.5) and (4.9), we choose t/c2=π−3, which makes even π2t/c2 numerically small rather than large. The result is shown in figure 4.
5. Small Reynolds number: a preliminary result
We now pose(5.1)where ω* is expected to be of order κ/ν times ω0. Defining(5.2)we seek ω* as the solution, for sufficiently small κ/ν, of the equation(5.3)Here L−1f denotes the unique periodic solution q, for given periodic f, of the problem (which could be written as Lq=f)(5.4a)(5.4b)under suitable hypotheses about f and q. In this section, henceforth we consider only t∈(0,c2] and limits as t↓0.
For existence of q, a sufficient hypothesis about the singularity of f at (z, t)=(0, 0) is that(5.5)or that f be bounded like the less critical part of u0.∇ω0. For uniqueness of q, a sufficient hypothesis about the singularity of q at (z, t)=(0, 0) is that(5.6a)(5.6b)or that q be bounded like the less critical part of ζ in theorem 5.1 below. The bounding functions in (5.5), (5.6a) and (5.6b) are relevant because in theorem 5.1 the worst parts of u0.∇ω0, ζ and ∇ζ behave, respectively, like , and near (z, t)=(0, 0). The bounding functions are slight generalizations of these (being larger as ρ↓0).
For periodic functions f that are suitably bounded (u0.∇ω0 is such a function), L−1f can be written in terms of the fundamental solutions E1 and E2 of the heat equation for one and two space dimensions, respectively. These are(whenceandOf course, we have met these functions already in (3.1), (4.1) and (4.5). We define(5.7a)where(5.7b)and let(5.8)Then(5.9)Alternatively,(5.10)It is not difficult to prove equality of these two formulae (each of which is useful) when f is periodic.
The following preliminary result is a first step towards reducing (5.3) to a contraction mapping of a small ball in a function space. Note that γ has the physical dimension of vorticity and that c−2t and η are non-dimensional.
Define and γ≔κ2/(192νc2). Then(5.11)where(5.12a)(5.12b)(5.12c)and A0, A1, A2 are absolute constants (numbers independent of the data).
The proof is long, intricate and tedious. For effective estimates of (u0.∇ω0)S, it seems necessary to decompose u0,S into the sum of u0,0, the velocity due to the diffusing vortex (n=0) at the origin; u0,1, the leading term near the origin of the irrotational velocity due to the other vortices (n≠0); u0,2, the remaining irrotational velocity due to the other vortices; and u0,3, the rotational velocity due to the other vortices. The vorticity ω0,S is decomposed into the sum of ω0,0, the vorticity due to the vortex at the origin, and ω0,1, the vorticity due to the other vortices. Then u0,0.∇ω0,0=0, as was explained in the penultimate sentence of §2 (although in the context of small time). For t↓0, the dominant part of (u0.∇ω0)S/ν is(5.13)this is the basis of the explicit term in (5.11).
The theorem is encouraging in two respects. First, ζ is indeed of order κ/ν times ω0, with a coefficient that is not large numerically unless the numbers Aj are huge. (I have calculated all the coefficients of functions bounding (u0.∇ω0)S, but to estimate the Aj would be difficult and unrewarding, because many terms of unknown sign are involved.) Second, ζ is smaller than ω0 not only by a factor of order κ/ν, but also by a factor of order t/c2.
The explicit term in (5.11), which dominates for t↓0 (it becomes −2γρ2 exp(−ρ2)sin 2θ in inner variables), confirms our expectation that the vorticity does not remain an even function of x and y (whereas all Stuart shear flows retain this symmetry). In fact, negative vorticity (clockwise vorticity) drifts into the first and third quadrants of S, at the expense of that in the second and fourth quadrants.
(a) On α as a reversed time
In retrospect, it seems that our interpretation of the parameter α as a reversed time (in §1, after equation (1.6)) was more fanciful than realistic, except for small t/c2 and small r/c if errors of 25% in the velocity can be tolerated. Here are some details to support this remark.
Presumably the solution of (2.2)–(2.4) tends to a parallel flow as t→∞; that is, presumably u(z, t)∼(u∞(y, t), 0) as t→∞ (for some u∞ independent of x). In that case, u.∇ω becomes negligible again and (4.9) applies once more, although probably with a larger O-term and with t measured from a different origin. On the other hand, the velocity of the Stuart flows becomes independent of α as α↓0 and the tanh profile (1.5) is approached. Moreover, we noted after (4.9) that the velocity profiles in (4.9) and (1.5) are close to each other not for a large value of t/c2 (as a good analogy would imply) but for a small value.
As was mentioned in §1, the vorticity of every flow of the Stuart family is an even function of x and of y; theorem 5.1 displays a loss of this symmetry, to second order for small Reynolds number. Since the convective rate of change u.∇ω forces this perturbation, the loss of symmetry seems likely to be stronger at higher Reynolds number.
Consider α↑1 (that is, consider Stuart flows departing only slightly from the classical vortex row in (1.1), corresponding to t↓0) and r↓0 (so that the dominant part of the velocity is not affected by the vortices at (2nc, 0), n≠0). This situation represents our best chance of a useful choice of α(t), for small t. Such Stuart flows have the velocity(6.1)where . This velocity is to resemble, as closely as possible, the velocity(6.2)of a single diffusing vortex. Thus, we require thatwhere ≈ denotes approximate equality in some coarse sense; equivalently, that(6.3)where . We choose δ2=1, hence(6.4)because then both sides of (6.3) equal ρ2+O(ρ4) as ρ↓0 and equal 1+O(ρ−2) as ρ→∞. However,(6.5)
(b) Distortion of the vorticity
If the time t/c2 and the Reynolds number κ/ν are both small, then theorems 3.1 and 5.1 yield the same asymptotic result, at least formally in the case of equation (3.4). In both cases, the forcing term is due to the velocity field with stream function(6.6)which is the velocity near the origin induced by the vortices not at the origin, acting on the vorticity gradient,(6.7)If the convective terms in (3.4) (those with coefficients β(ρ) and γ(ρ)) are neglected, there results(6.8)which agrees with the ζ in (5.11) for t↓0. A referee has kindly described this perturbation as the tendency of the vorticity to align with the strain induced by the sum of the other vortices.
(c) Remaining work
Evidently, much remains to be done before the objectives (3.3) and (5.1) become results, with ω* and ω* suitably bounded. However, the inequalities used to prove theorems 3.1 and 5.1, together with the additional a priori bound (in an analogous problem) that was mentioned after theorem 3.1, augur well for this remaining work.
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