The classical plane Couette flow, plane Poiseuille flow, and pipe Poiseuille flow share some universal 3D steady coherent structure in the form of "streak-roll-critical layer". As the Reynolds number approaches infinity, the steady coherent structure approaches a 3D limiting shear of the form ($U(y,z), 0, 0$) in velocity variables. All such 3D shears are steady states of the 3D Euler equations. This raises the importance of investigating the stability of such inviscid 3D shears in contrast to the classical Rayleigh theory of inviscid 2D shears. Several general criteria of stability for such inviscid 3D shears are derived. In the Appendix, an argument is given to show that a 2D limiting shear can only be the classical laminar shear.
Deep Dive into Stability Criteria of 3D Inviscid Shears.
The classical plane Couette flow, plane Poiseuille flow, and pipe Poiseuille flow share some universal 3D steady coherent structure in the form of “streak-roll-critical layer”. As the Reynolds number approaches infinity, the steady coherent structure approaches a 3D limiting shear of the form ($U(y,z), 0, 0$) in velocity variables. All such 3D shears are steady states of the 3D Euler equations. This raises the importance of investigating the stability of such inviscid 3D shears in contrast to the classical Rayleigh theory of inviscid 2D shears. Several general criteria of stability for such inviscid 3D shears are derived. In the Appendix, an argument is given to show that a 2D limiting shear can only be the classical laminar shear.
There has been a lot of continuing interest in searching for 3D steady solutions (or traveling wave solutions in a different frame) in plane Couette flow, plane Poiseuille flow, and pipe Poiseuille flow [17] [16] [21] [11] [22] [20] [6] [4] [7]. There seems to be confirmation of their existence in experiments [7]. Recent numerical studies of [22] [21] [20] reveal that the so-called lower branch steady states in the plane Couette flow, plane Poiseuille flow, and pipe Poiseuille flow share some universal steady coherent structure in the form of "streak-roll-critical layer". As the Reynolds number approaches infinity, the steady coherent structure approaches a 3D limiting shear of the form (U (y, z), 0, 0) in velocity variables. All the 3D shears of this form are steady states of the 3D Euler equations. This raises two important questions: (1). What is the special property of the limiting shear? (2). What is the nature of stability of 3D inviscid shears in contrast to the classical Rayleigh theory of 2D inviscid shears? The first question was addressed in [14]. It turns out that the limiting shear satisfies a necessary condition: ∆U f (U )dydz = 0 for any function f . We shall address the second question in this study. We shall use the channel flow (plane Couette flow and plane Poiseuille flow) as the example.
As the Reynolds number decreases from infinity, the limiting 3D shear as a steady state deforms into the lower branch steady state; while the 3D shear itself undergoes a slow drifting toward the classical laminar shear. In fact, all the shears (3D and 2D) form a stable submanifold of the classical laminar shear. These shears can play a fundamental role in the transition to turbulence from the classical laminar shear [13].
The inviscid channel flow is governed by the 3D Euler equations (2.1)
where (u 1 , u 2 , u 3 ) are the three components of the fluid velocity along (x, y, z) directions, and p is the pressure. The boundary condition is the so-called slip condition
where a < b, and u i (i = 1, 2, 3) are periodic in x and z directions with periods ℓ 1 and ℓ 3 .
We start with the steady shear solutions of the 3D Euler equations:
where U (y, z) is periodic in z with period ℓ 3 . Linearize the 3D Euler equations with the notations
where k is a real constant and c is a complex constant, we obtain the linearized 3D Euler equations
Two forms of simplified systems can be derived:
with boundary condition v(a, z) = v(b, z) = 0 and v, w are periodic in z; and
with boundary condition ∂ y p(a, z) = ∂ y p(b, z) = 0 and p is periodic in z. We are not successful in utilizing the system (2.7)-(2.8). System (2.9) turns out to be fruitful. The first result that can be derived from system (2.9) is the Howard semi-circle theorem.
Theorem 2.1. [8] [3] The unstable eigenvalues (if exist) lie inside the semi-circle in the complex plane:
where M = max y,z U , and m = min y,z U .
Proof. Multiply (2.9) with p, integrate by parts, and split into real and imaginary parts; we obtain that
where
Expand this inequality and utilize (2.10)-(2.11), we arrive at the semi-circle inequality in the theorem.
Our next goal is to find a counterpart of the Rayleigh criterion [2]. For that goal, we need to introduce the transform Theorem 2.2. For U (y, z) satisfying the constraint
, where M = max y,z U ,
, where M = max y,z U , and m = min y,z U .
Proof. Multiply (2.13) with p, integrate by parts, and split into real and imaginary parts; we obtain that
Equation (2.18) directly implies the first claim in the theorem. The second claim is along the spirit of the Fjortoft theorem [2]. Multiply (2.18) by c r and add (2.17), we obtain the second claim.
Next we will derive a relation between c i and k.
Proof. Multiply (2.13) with p and integrate by parts, we obtain that
The left hand side of (2. 19) is less than or equal to
The right hand side of (2. 19) is greater than or equal to
which leads to the claim of the theorem.
It is obvious that Theorems 2.2 and 2.3 apply to the 2D shears U (y) too. Theorem 2.2 is not the exact 3D counterpart of the 2D Rayleigh criterion. The exact counterpart seems elusive. Next we will derive a variation formula for the unstable eigenvalue. This type of formulas was initially derived by Tollmien [19] [15] for 2D shears. They are useful in deriving unstable eigenvalues near neutral eigenvalues. For 3D shears in atmosphere problems [1], specific approximations can make the stability problem very similar to the 2D Rayleigh problem. In such a case, a similar variation formula can also be derived to predict unstable eigenvalues near neutral eigenvalues [1]. In our current case, no approximation can be made, and we have a much harder problem. We have to work with the pressure variable of which the singularity nature is not clear even for 2D shears. We can derive a formula near an unstable eigenvalue, but its limit to a neutral eigenvalue is elusive and finding a neutral eigenvalue here is more challengi
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