Plane shearing waves of arbitrary form: exact solutions of the Navier-Stokes equations

We present exact solutions of the incompressible Navier-Stokes equations in a background linear shear flow. The method of construction is based on Kelvin's investigations into linearized disturbances in an unbounded Couette flow. We obtain explicit formulae for all three components of a Kelvin mode in terms of elementary functions. We then prove that Kelvin modes with parallel (though time-dependent) wave vectors can be superposed to construct the most general plane transverse shearing wave. An explicit solution is given, with any specified initial orientation, profile and polarization structure, with either unbounded or shear-periodic boundary conditions.

In 1986 Craik and Criminale [1] presented a class of exact solutions of the Navier-Stokes equations which were wavelike disturbances in background shear flows. Since then these solutions have proved extremely useful in the study of astrophysical and atmospheric fluid dynamics; a very useful collection of exact solutions can be found in [2]. The approach taken in [1] was a generalization of a century-old method invented by Kelvin [3] to study linearized perturbations of Couette flows. These shearing wave solutions, also referred to as Kelvin modes, have time-dependent wavevectors and amplitudes. Although a single Kelvin mode is an exact solution of the full Navier-Stokes (NS) equations, it has been remarked [1] that until about 1965 there seems to be no evidence that this was so recognized; in fact, the first published mention is as late as 1983 [4]. Moreover, an explicit formula has been published [1,3] for only one of the three components of the disturbance.
In this Letter we present exact solutions for all three components of the velocity field of a Kelvin mode, in closed form using only elementary mathematical functions. We identify a subset of these modes whose wavevectors -though time-dependent -remain parallel to each other for all time. These are used to synthesize the most general plane transverse shearing wave, which can have any specified initial orientation, profile and polarization structure, with either unbounded or shear-periodic boundary conditions. Let (ê 1 ,ê 2 ,ê 3 ) be the unit basis vectors of a Cartesian coordinate system in the laboratory frame. Using notation x = (x 1 , x 2 , x 3 ) for the position vector and t for time, we write the total fluid velocity as (Sx 1ê2 + v), where S is the rate of shear parameter and v(x, t) is the incompressible disturbance (∇·v = 0) which obeys the NS equations: We seek a solution in the form of a single Kelvin mode v k (x, t) = Re A(k, t) exp i k sh (t)·x , where the time-dependent sheared wavevector, k sh (t), has components with k ≡ (k 1 , k 2 , k 3 ) being a constant wavevector. Our task now is to determine the amplitudes A(k, t). Incompressibility requires that k sh (t)·A(k, t) = 0 . Therefore, when eqns. (2) and (3) are substituted in eqns. (1), the nonlinear term, (v·∇) v vanishes because The pressure can be eliminated by using the second of eqns.
We now obtain explicit solutions for A. To do this, define a new amplitude variable, a(k, t), where G ν (k, t) is a Fourier-space viscous Green's function, When eqns. (6) and (7) substituted in eqn. (5), we obtain the following equations for the three components of a(k, t) : which expression is given in [3]. When this is substituted in eqns. (9) and (10), the latter can be integrated to obtain expressions for a 2 (k, t) and a 3 (k, t). However, neither Kelvin nor anyone else, to the best of our knowledge, have published explicit formulae for these two components. Thus we were pleasantly surprised to find that a 2 (k, t) and a 3 (k, t) could be expressed entirely in terms of elementary functions: Incompressibility requires that k sh (t)·a(k, t) = 0, which is guaranteed if the initial conditions are chosen such that k·a(k, 0) = 0 . When eqns. (6), (7), (11) and (12) are substituted in eqn. (2), we obtain the full velocity field of a single Kelvin mode. Until now we have considered an unbounded flow. However, in numerical simulations of the local dynamics of differentially rotating discs in astrophysical systems [5,6], it is customary to employ "shear-periodic" boundary conditions. Let us define sheared coordinates by These may be thought of as the Lagrangian coordinates of fluid elements that are carried along by the background shear flow. A function is said to be shear-periodic when it is a periodic function of (x sh 1 , x sh 2 , x sh 3 ) with periodicities (L 1 , L 2 , L 3 ), respectively. The phase of the function v k can be written as k sh (t)·x = k·x sh . Therefore, a shear-periodic Kelvin mode has wavevectors k ∈ (2πm 1 /L 1 , 2πm 2 /L 2 , 2πm 3 /L 3 ), where the m i take any integer values.
We now use the explicit expressions obtained for the Kelvin modes to construct the most general plane transverse shearing wave. Let us consider two Kelvin modes, v k (x, t) and v k ′ (x, t) corresponding to wavevectors k and k ′ , which are parallel to each other but could differ in magnitudes. Using eqns. (3), we see that the corresponding sheared wavevectors, k sh (t) and k ′ sh (t), are also parallel to each other for all time. Incompressibility implies that v k (x, t) and v k ′ (x, t) are perpendicular to k sh (t) and k Let us choose a unit vectorn = (n 1 , n 2 , n 3 ), and define the sheared (non-unit) vector n sh (t) by Superposing all Kelvin modes with wavevectors q = qn, where −∞ < q < ∞, we obtain an exact plane-wave solution of the NS equations with wavefronts perpendicular to n sh (t): where the dimensionless and scale-invariant functions, F i (Q), are defined by For shear-periodic boundary conditions, the integral over q in eqn. (15) should be replaced by an appropriate sum. The W (q) are Fourier-space initial conditions corresponding to the a(k, 0) of eqns. (11) and (12), and must satisfy the incompressibility condition,n· W (q) = 0 . They are determined by the initial profile and polarization stucture of the plane wave.
At t = 0, the wavefronts are perpendicular ton, so we write v(x, 0) = W (n·x), wherê n·W = 0. Note that the only constraint on the initial condition, W , is that it is a vector field that is perpendicular everywhere to the unit vectorn; otherwise it is a quite arbitrary function of its one argument. Thus, no restriction need be placed on the initial profile and polarization structure of the initial conditions. Given W (y), we can determine , and use this in eqn. (15) to calculate v(x, t). Eqn. (15) is a complete solution for a general plane shearing wave, expressed in terms of a Fourier integral. However, it is physically more transparent to rewrite the right side in terms of real-space quantities. To do this, we must introduce the real-space viscous Green's function, whose natural definition is with respect to the sheared coordinates [7]: The properties of this function are discussed in [7,8], where it is shown that it takes the form of a sheared heat kernel, which is an anisotropic Gaussian function of x sh with timedependent coefficients; all the principal axes increase without bound and rotate against the direction of the background shear. Noting that n sh (t)·x =n·x sh , we can write the general form of the plane shearing wave as As an illustrative example let us consider the following initial condition, corresponding to a polarized wavepacket with wavevector pointing along the x 2 -axis:n =ê 2 , W 1 ( v(x, t) evaluated explicitly. We present the results graphically in Fig.(1) for two cases, one linearly polarized and the other right circularly polarized. As the wavepackets are sheared, they undergo transient amplification due to the combined action of shear and viscosity, and at late times suffer viscous damping.
In conclusion, we have constructed exact solutions of the Navier-Stokes equation with a background linear shear flow. All three components of the velocity field of the Kelvin modes are given in closed form using only elementary mathematical functions. It is demonstrated that, when Kelvin modes with parallel wavevectors are superposed, they remain exact solutions. We give in explicit form the most general plane transverse shearing waves, with any specified initial orientation, profile and polarization structure, with either unbounded or shear-periodic boundary conditions. These represent the local structure of any disturbance in general shear flows, and can therefore be expected to find many applications in the theory and simulations of astrophysical and atmospheric flows.