The start-up and linear growth phases of the Richtmyer-Meshkov instability have been in- vestigated using a novel shock-resolved Navier-Stokes simulation. By assuming a linearized form for the solution with a perturbed interface, where the base flow is the solution to the nonlinear 1-D Navier-Stokes equations for a shock incident on a density interface, the
problem could be reduced to a 1-D computational domain. To efficiently resolve the full structure of the shocks, the interface closures of Chapter 2 were employed in a refinement scheme that tracked the solution features based on the inviscid Riemann solution. A full set of validation tests for the explicit fourth-order code has shown that the method is time stable and converges at the expected fourth-order rate.
Results from the simulations show that for weak and intermediate strength shocks, the impulsive model is adequate for prediction of the asymptotic growth rate of the perturbation.
For strong shocks, the model from Wouchuk (2001a) accurately predicts the asymptotic growth rate. Transient growth shows general agreement with Wouchuk’s model for weaker shocks, while for stronger shocks the same model significantly underestimates the amplitude of oscillation in the growth rate. The characteristic start-up time proposed by Lombardini (2008) is verified as a good estimate of the time for the instability to enter the linear growth regime. The extent of the influence of viscosity is predicted by a Reynolds number based on the perturbation wave number. Existing models of viscous attenuation are found to poorly predict the growth rate of the instability in most cases.
Further investigation within the current formulation may begin with a broader explo- ration of the parameter space, primarily in MI and Atwood number. Of great interest would be fully resolved simulations of high-MI shocks, as the existing theory and body of simulation results is somewhat limited in this area. The greatest hurdle for these problems is the resolution requirements, as discovered for the MI = 2.20, A = 0.6 case, where suc- cessively greater refinements are required to resolve the very steep gradients encountered during impact of the shock with the interface. Computational cost may also be reduced with more sophisticated time-stepping routines, where the stability boundaries permit larger CFL numbers for the locally refined grids.
Extension of the present approach should begin with an implementation of the two- species gas mixture case, which is of interest because the relative values ofγ0 andγ00 have a strong influence on the transient growth of the solution. Comparison to experimental results would be improved, as each fluid could be modeled more accurately. Further work may also be done to progressively weaken the approximations made in the current formulation, depending on the particular focus of the future investigation.
Appendix A
Analysis of Finite-Difference Schemes
A.1 GKS Stability Theory
Stability of a discrete approximation to a linear hyperbolic partial differential equation may be analyzed for the Cauchy problem on an infinite domain using standard Fourier techniques (e.g., Lomax et al., 2001, §7.7), but on finite domains with imposed boundary values, this analysis is not appropriate and GKS stability theory, after Gustafsson et al. (1972), must be used. This discussion largely follows that of Carpenter et al. (1993), and is presented here as the precursor to the introduction of summation-by-parts operators, which are themselves shown to be GKS stable in Section 2.2.5.
The power of the GKS theory is that it allows separation of the stability problem into an infinite-domain Cauchy problem and two boundary problems. A necessary and sufficient condition for stability then requires that each problem have no eigensolution.
This is formulated for the discrete problem (here following Berger, 1985) by assuming a normal mode form for the solution vector
u(j∆x, n∆t)≈unj =znφj, (A1.1) whereφj is the solution to the resolvent equation for the finite difference scheme (this form comes from a discrete Laplace transform in time that underpins the GKS analysis). Upon assuming a form φj = φ0κj, the resolvent equation will give a polynomial in κj that has
multiple roots κjk, so we can write down the general form φj = X
|κs|≤1
p(j, s)κjs(z), (A1.2)
wherep is the polynomial coefficient inj of order one less than the multiplicity of the root κs. Stability, in the sense that the solution (A1.1) is bounded, then requires that there be no solutions forφj with
1. |z|>1 and |κ|<1, or 2. |z|= 1 and |κ| ≤1.
These conditions bound φj by excluding terms from the summation equation (A1.2) that could cause growth: (1), where |z| >1, or (2), in the degenerate case where |z|= 1. The remaining combinations of |z|and|κ|leave equation (A1.1) bounded in space and time.
The GKS theory may also be extended to the semidiscrete problem if a Runge-Kutta time-marching scheme is used with a discretization that is inside the stability region. The results look similar, but it is useful to cast the problem in the semidiscrete form to com- pare the resulting stability conditions to those required for asymptotic stability. Following Carpenter et al. (1993), we assume a normal-mode solution to the semidiscrete problem described by (2.2.6)
uj(t) = eGtφj, (A1.3)
whereG= diag(λA) are the eigenvalues of A. Again we assume a form for φj
φj =φ0κj. (A1.4)
An eigensolution to the initial-boundary value problem (2.2.6) is then defined as a nontrivial function with<(G)≥0 that satisfies
1. for<(G)>0,|κ|<1 (so a GKS-eigenvalue exists), or
2. for <(G) = 0 and |κ| = 1, a perturbation to |κ| = 1− with > 0 generates an eigenvalue<(G)> δ, where δ >0 (i.e., a generalized GKS eigenvalue exists).
The condition for stability is then that there exists no such eigensolution; see that (1) allows a solution bounded in x that would grow exponentially in time, and that (2) covers
the limiting case. For a given scheme, stability is determined by assuming a solution of the normal mode form (A1.3) and showing that the resulting κ is not a GKS eigenvalue.
To demonstrate the resolvent condition and the form of the roots κ, we use an example from Carpenter et al. (1993) and start with the generalized eigenvalue equation derived from equation (2.2.6):
Guj =Auj. (A1.5)
The resolvent equation follows upon substitution of equation (A1.3); for the fourth-order implicit scheme, this has the form
1
4φj−1+φj +1 4φj+1
G= 3
4(φj+1−φj−1). (A1.6) Substituting for φj by (A1.4), we obtain a quadratic equation inκ
1
κ + 4 +κ
G= 3
κ− 1 κ
. (A1.7)
The first root (κ1 =κ) is obvious, but to find the second, substitute a function f(κ) such that
f2+ 4f+ 1
G= 3(f2−1), (A1.8)
where f2+ 4f + 1 = (κ2 + 4κ+ 1)g(κ) and f2−1 = (κ2 −1)g(κ), for some amplitude functiong(κ). Doing the algebra, we find a quadratic inf(κ) with the two roots we seek,
κ1=κ and κ2 = −κ−2 2κ+ 1, soφj has the general form
φj =C1κj1+C2κj2, (A1.9) whereC1andC2are constants determined from the boundary treatment. Note that|κ1| ≥1 implies |κ2| ≤1 here; Gustafsson et al. (1972) shows that in the discrete case with |z|>1, there will be N l linearly independent solutions with |κ| < 1 and N r with |κ| > 1 for a difference stencil with l points to the left and r to the right (N is the size of the system, which is clearly assumed to be uniform).
A final quirk of Lax/GKS stability is that the eigenvalues of the spatial finite-difference
scheme must satisfy
<(λj)≤ω, ω≥0
as a necessary condition. This means that the eigenvalues asymptotically approach a bound in the right-hand complex plane as N → ∞. This may appear somewhat alarming at first glance for time stability, but notice that the eigenvalues of a centered difference scheme on an infinite domain (as approached in the limit N → ∞) have zero real part; the key is to keepω = 0 as the limit when finite boundaries are present.
Regarding GKS stability at the interfaces, it is prohibitively difficult to perform this analysis directly on stencils as large and complex as those used in the interface schemes, so instead we rely on the summation-by-parts property to guarantee GKS stability.