The most important difference between the inviscid dispersion relation for the 2D analysis (4.55) and the inviscid dispersion relation for the 1D analysis (4.58), is the appearance of the factor 1/k inside the square root. This means that according to the 2D analysis, the velocity difference for which the wave is unstable is dependent on the wavelength of the wave (compare to (4.17)):
∆U2= (uG− uL)2> tanh(khint) ρL +tanh(k(H − hint)) ρG (ρL− ρG) g cos φ k . (4.59)
For a given velocity difference, the system can be unstable to waves with short wavelengths (large k) while being stable for waves with long wavelengths (small k). However, taking the limit of long wavelengths was seen to remove the dependence on k (4.58). According to Montini (2011) [131], ‘the logical consequence is that the averaging of the equations in order to obtain a one-dimensional system intrinsically implies the long wavelength (k → 0) assumption’. The two-fluid model assumes negligible vertical velocities and a requirement for this is that the interfacial height does not vary too steeply along the horizontal direction. This is assured with the long wavelength assumption. The long wavelength assumption can be associated with the assumption L H, discussed in subsection 2.3.1 and subsection 3.4.2.
The amplitudes of the velocity perturbations, similar to (4.52), are calculated by determining the coefficients BG and BL in (4.54) using equations (4.43). The coefficients are given by
BG = −i kuG− ω k sinh (k(H − hint)) ˆ η, (4.60a) BL= i kuL− ω k sinh (khint) ˆ η. (4.60b)
According to the notation of subsection 4.3.2 we may write η = ∆hint. Therefore the horizontal velocity
perturbations can be calculated to be
∆uG= ∂φ0G ∂s → ∆ˆuG = (kuG− ω) cosh (k(H − hint− z)) sinh (k(H − hint)) ∆ˆhint, (4.61a) ∆uL= ∂φ0L ∂s → ∆ˆuL= − (kuL− ω) cosh (k(z + hint)) sinh (khint) ∆ˆhint. (4.61b)
The vertical velocity perturbations, which are superimposed on the base flow of wG = wL= 0, are given
by ∆wG= ∂φ0G ∂z → ∆ ˆwG = i (kuG− ω) sinh (k(H − hint− z)) sinh (k(H − hint)) ∆ˆhint, (4.62a) ∆wL= ∂φ0L ∂z → ∆ ˆwL = i (kuL− ω) sinh (k(z + hint)) sinh (khint) ∆ˆhint. (4.62b)
Again, the vertical coordinate used here is related to h via z = h − hint.
4.5
Comparison of dispersion relations
4.5 Comparison of dispersion relations 49
• for 1D flow (two-fluid model) the dispersion relation (4.14) with perturbations (4.35) and (4.36), • for unbounded 2D flow (Kelvin-Helmholtz) the dispersion relation (4.48) with the velocity perturb-
ations (4.52) and (4.53),
• and for bounded 2D flow the dispersion relation (4.55) and the perturbations (4.61) and (4.62). We plot the dispersion relations in Figure 4.2, using the values in Table 4.1. Apart from the velocities, these are the test case parameters used in section 5.2. The stability limit for the two-fluid model given by (4.17) is uG− uL= 7.55 m/s.
Table 4.1: Kelvin-Helmholtz test case parameters.
uL ρL ρG g H φ hint/H
0.5 m/s 998 kg/m3 1.2 kg/m3 9.81 m/s2 0.01 m 0 rad 0.3
(a) (b)
(c) (d)
Figure 4.2: Inviscid dispersion relations, using the parameters given in Table 4.1.
In the figure the limiting behavior of the dispersion relations is readily observed. The dispersion relations for the 1D bounded and 2D bounded cases indeed converge for long wavelengths, and the dispersion relations for the 2D unbounded and 2D bounded cases converge for short wavelengths. The 2D cases are always unstable to perturbations with the shortest wavelengths. Only if viscosity were to be added stable solutions can be reached. The instability of the inviscid 2D case does not mean that a
similar case that would include viscous terms is ill-posed, since the analysis is based on the linearized equations, and not on the full set. Moreover, the viscous terms in the 2D case contain derivatives of the velocity and thus should be included in the homogeneous part of the equations when analyzing viscous flow.
The inviscid 1D bounded model is either stable and well-posed or unstable and ill-posed for all wavelengths, depending on the velocity difference. The inviscid equations form the homogeneous part of the viscous equations, so that the viscous two-fluid model can also directly said to be ill-posed.
For a slightly differenct casein which all models have a nonzero imaginary part of ω, Figure 4.3 shows the growth of initial perturbations of a certain wavelength with time. Depending on the sign of the imaginary part of the ω to which the perturbations correspond (referring to the ± in (4.55)), the perturbation is amplified or damped. The 2D Gerris simulations match the predicted growth rate initially, but deviate after some time. This is to be expected; nonlinear effects are not taken into account in the linear stability theory.
The two-fluid Rosa simulations stick to their predicted growth rate for longer. This is to be expected since this growth rate can be deduced directly from the full set of inviscid equations, not from a linearized version as is the case for the 2D equations. For the inviscid case, the linear stability analysis performed for the two-fluid model in section 4.4 yields the same dispersion relations as the treatment in subsection 4.3.1 which is based on the full, non-linearized equations. These two-fluid simulations are fragile though, since the model equations are ill-posed, and if there is not enough numerical diffusion small wavelength perturbations might grow with an unbounded growth rate (see the discussion in subsection 4.3.3). Nevertheless, these plots indicate the correctness of the simulation codes, and our understanding of linear stability theory.
(a) Initialized with a perturbation correspond- ing to the plus sign in (4.55).
(b) Initialized with a perturbation correspond- ing to the minus sign in (4.55).
Figure 4.3: Comparison of the growth rate of the perturbations in Gerris simulations, in 2D linear stability theory (4.55), in Rosa (two-fluid) simulations, and in 1D two-fluid model analysis (4.14), all inviscid. Using most parameters given in Table 4.1 but with ρG= 980 kg/m3 and uL= 0.01 m/s, uG= 0.06 m/s. The wavelength is
equal to the domain height and length, 0.1 m.