Comparison to previous work

The stability problem (3.3) can be understood in the context of previous work. Benjamin [3] and Yih [34] have studied the stability of fluid flowing down an inclined plane with a free surface in two dimensions, and Kao [12, 13] has extended this by considering the effects of a stratified two-fluid system flowing down an inclined plane, although he did not consider arbitrary density profiles. In addition Yih has studied the stability of flow with viscous stratification [35] in two dimensions, which although is quite different physically from the present study bears many mathematical similarities to it.

Finally, Sangster [23] studied the problem of the flow of a two-fluid system down an inclined surface with a rigid lid (no free-surface) in two dimensions, with and without surface tension at the fluid-fluid interface. Therefore, the present study corresponds to the problem studied by Sangster for an inclination of 90◦, zero surface-tension, and the addition that one of the boundaries is towed at a fixed rate. Sangster did not investigate the time-evolution of the thickness of one stream relative to the other, but only considered the stability analysis for a fixed ratio of the bottom fluid thickness to top fluid thickness. Through the use of a long-wave expansion (as used here) in addition to a truncated Frobenius series for the eigenfunction (not used here), Sangster obtained approximations to the real and imaginary components of the eigenvaluec. The analysis

of Sangster was limited to the condition that the thickness of the lower stratum be greater than that of the upper, which is exactly the opposite of the situation of primary attention in the current investigation. In this thesis, primary attention is given to the situation in which the size of the entrained layer (corresponding to the lower-stratum in Sangster’s study) is small, although much of the analysis presented here does not require this condition. Finally, Sangster’s calculations are rather laborious and the resulting eigenvalue expressions are not shown to simplify as do the calculations presented here, which result in the comparatively compact expressions (3.15) and (3.20). As will be seen in the axisymmetric problem, the corresponding expressions do not simplify to such compact expressions.

In the investigations of Yih [34] and Kao [12, 13], there were found “hidden neutral modes” which are analogous to the modes found here. These modes are termed hidden since they arise from the misalignment of isopycnals with gravity in [3, 34, 12, 13], and the viscous stratification in [35], and they vanish if these physical conditions are not present, for instance in homogeneous flows. Further the modes are neutral to leading order for long waves. Thus the modes found in the present study lie under this classification of hidden neutral modes.

In order to understand the connections to these previous investigations, we consider a two-dimensional stratified fluid flow bounded by two walls of infinite extent which enclose a channel (no free surface present) that is inclined with respect to gravity by an angle θ. One wall of the channel is towed at speed U0 which may be zero, and the

density profile is arbitrary. See figure (3.3) for a diagram.

We let x = (x1, x2) be the horizontal and vertical coordinates with respect to the

channel walls and letu= (u1, u2) be the corresponding velocities. The nondimensional

Boussinesq approximation equations are given by

ReD Dtu = −∇p+ ∆u+ Re F r2 ρ (sinθ,−cosθ) ∇ ·u = 0 D Dtρ = 0,

We allow for an arbitrary density profile varying only in the vertical direction ρB(x2),

 

U₀

x1,u1 x2,u2

Figure 3.3: Diagram of stratified flow through a channel inclined with respect to gravity.

velocity are related by

U00(x2) +

Re

F r2 sinθ ρB(x2) +β = 0 (3.22)

where as before β is a constant arising for the horizontal pressure gradient. A streamfunction ψ exists with relation to the velocities

u1 = − ∂ψ ∂x2 u2 = ∂ψ ∂x1

The streamfunction formulation of the Boussinesq approximation is

Re∂ ∂t(∆ψ)−Re ∂ψ ∂x2 ∂ ∂x1 (∆ψ) +Re∂ψ ∂x1 ∂ ∂x2 (∆ψ) = ∆2ψ− Re F r2 cosθ ∂ρ ∂x1 + sinθ ∂ρ ∂x2

We allow for perturbation around the background solutions

ψ(x) = ψB(x) +εψ(x)eˆ ik(z−ct) ρ(x) = ρB(x) +ερ(x)eˆ ik(z−ct)

from which we derive the stability equation for streamfunction and density perturba- tions

(D2−k2)2ψ−i k Re (UB−c)(D2−k2)ψ−UB00ψ

− Re

F r2 (ikcosθ ρˆ+ sinθ Dρ) = 0ˆ

ˆ ρ=− ρ 0 B UB−c ˆ ψ

whereD =d/dx2. Substitution of ˆρgives

(D2−k2)2ψ−i k Re (UB−c)(D2−k2)ψ−UB00ψ +ikcosθ Re F r2 ρ0_B UB−c ψ+ sinθ Re F r2D ρ0_B UB−c ψ = 0 (3.23)

To the knowledge of the author, this stability operator for arbitrary background density profiles has not been previously documented. The previous investigation [3, 34, 12, 13, 35] have focused on the special case of a two-fluid system and thus the effects of the terms involving ρB in (3.23) entered analysis in the boundary conditions at the fluid-fluid interface only.

We now give an understanding of each component of the stability operator (3.23) in light of previous work and the present study. The top line of the operator is simply the classic Orr-Somerfeld equation for homogeneous fluid flow and does not give rise to neutral modes at leading order in the long-wave expansion. Indeed all modes are damped for sufficiently long waves or low Reynolds for the Orr-Somerfeld equation.

The term multiplying cosθarises from the horizontal component of density layering, and can be understood as part of the Taylor-Goldstein equation for the inviscid limit. Indeed the term ρ0_B/F r2 _{is the local Richardson number for the inviscid Boussinesq}

equation exactly. Therefore, we will refer to this term multiplying cosθ as the Taylor- Goldstein term. Notice that the wavenumber k multiplies the Taylor-Goldstein term and so it does not enter at the leading order of the long-wave expansion and therefore cannot give rise to leading order neutral modes. Therefore, for example in the case of θ= 0 all modes will be damped for sufficiently long waves.

The term multiplying sinθarises from the vertical component of the density layering and we will refer to it as the vertical density layering term. Since this term is not multiplied by the wavenumber, it enters the leading order equation for the long-wave expansion, and further since the eigenvalue c is present in this term, it allows for c= O(1) to leading order in this expansion. Thus this term is necessary in (3.23) for the presence of hidden neutral modes and it is the presence of this term in the stability operator that is the focus of the current study.

For further simplification we can substitute the relationship between the background density and velocity profiles

sinθ Re F r2ρ

0

B(x2) = −UB000(x2)

into (3.23) to obtain a stability operator in which we have eliminated ρB

(D2−k2)2ψ−i k Re (UB−c)(D2−k2)ψ−UB00ψ −ikcotθ U 000 B UB−c ψ−D U_B000 UB−c ψ = 0 (3.24)

The previous investigations [3, 34, 12, 13] have only studied the effects of the vertical density layering term in conjunction with the Taylor-Goldstein term, and once again these terms only entered in the boundary conditions at the fluid-fluid interface. In the Orr-Somerfeld equation the first power of k only enters in product with Re, while in the Taylor-Goldstein term k is not in product with Re. Therefore, in the long-wave expansion of c, the first correction will be a function of Re if the Taylor-Goldstein term is present and therefore there is the potential for a stability transition to occur as the Reynolds number is varied, as has been seen in [3, 34, 12, 13]. In the current study however, the vertical density layering term has been isolated so that there is no Taylor-Goldstein term present in the stability operator. In this case, the first power

of k only enters the operator in product with Re and so the first correction to c has trivial dependence on Re as it is only a scaling factor in the magnitude of the first correction, as seen in (3.12), (3.13), and (3.20). Therefore, the flow configuration under current investigation in which the density layering is completely vertical has the interesting feature that its stability or instability to long waves is independent of the Reynolds number, and only depends on the density profile. Of course the magnitude of the growth or decay rates of disturbances does depend on the Reynolds number. This interesting feature was also found by Yih [35] in the stability analysis of a flow with viscous stratification.

We point out a few notes regarding (3.24). First, the simplification (3.24) is not valid forθ = 0 since in this case we must haveU_B000 ≡0. Next, In the case thatθ =π/2 so that the channel is oriented vertically, we obtain the stability operator (3.3). Technically, for this comparison we would need to make the substitutions x2 →x, x1 → −z, UB → −wB, k → −k, c→ −c, since thez-direction is opposite of thex1-direction forθ =π/2.

However, by two fortuitous sign cancellations the substitution may be made naively with the correct stability operator obtained.

Finally, ifc=UBsomewhere in the flow then the stability operator (3.23) is singular due to both the Taylor-Goldstein term and the vertical density layering term, and further the vertical density layering term promotes the degree of the singularity. In the generic case in which U_B0 (x2) 6= 0 at the point that c = UB, then the singularity is promoted by the vertical density layering term from first order to second order, and therefore the singularity remains regular. However in the inviscid limit of (3.23), the Taylor-Goldstein equation has only a regular singular point while the vertical density layering term creates an irregular singular point. It should be noted however that viscosity is necessary for the existence of the background solution (3.22) if θ 6= 0