Numerical Algorithm (Direct Numerical Simulation)
A highly
accurate algorithm has been developed to study the process of spatial transition
to turbulence. The algorithmic details of the direct numerical simulation of
transition to turbulence in a boundary layer based on a formulation in terms of
vertical velocity and vertical vorticity are presented. Issues concerning the
boundary conditions are discussed. The linear viscous terms are discretized
using an implicit Crank-Nicholson scheme, and a low-storage Runge-Kutta method
is used for the nonlinear terms. For the spatial discretization, fourth-order
compact finite differences have been used, as these have been found to have
better resolution compared to explicit differencing schemes of comparable order.
The number of grid points that are needed per wavelength is close to the
theoretical optimum for any numerical scheme. The resulting time-discretized
fourth-order equations are split up into two second-order equations, resulting
in Helmholtz- and Poisson-type equations. The boundary conditions for the
Laplacian of the vertical velocity are determined using an influence matrix
method. A robust multigrid algorithm has been developed to solve the resulting
anisotropic elliptical equations. For the outflow boundary, a buffer domain
method, which smoothly reduces the disturbances to zero, in conjunction with
parabolization of the Navier-Stokes equations has been used. The validation of
the results for the DNS solver is done both for linear and weakly nonlinear
cases.
High Performance Computing
The spatial transition to
turbulence has been simulated using a parallel multigrid strategy. The multigrid
has been parallelized using a domain decomposition technique. Linear
interpolation, full weight for restriction, and line Gauss-Seidel for smoothing
has been used. The parallel computing performance characteristics, speedup and
scaled efficiency have been determined. Both the algorithmic and implementation
scalability features have resulted in efficient parallelization.
Message-passing-interface (MPI) has been used for the communication between the
various nodes. The K-mechanism of laminar breakdown has been simulated using
direct numerical simulation (DNS) of the Navier-Stokes equation. The typical
structures of transition, high shear layer, $\Lambda$-vortex, spikes, and
longitudinal vortices have been observed.
Events of Transition (Physics of the flow)
Spatial transition
to turbulence for a flat-plate boundary-layer has been simulated using direct
numerical simulation. For the case of forced transition, a single frequency
disturbance has been introduced, and fundamental breakdown and the
characteristic events of a transitional boundary layer have been observed. The
main events and structures are the inflectional nature of the disturbance
velocity profiles as well as the mean velocity. The presence of a
$\Lambda$-vortex, significance of streamwise vorticity, spanwise vorticity,
shear layer, and speed of propagation of these structures are discussed in
detail as well as their role in the transition process and in the process of
breakdown to turbulence.
Resonance theory for transition to turbulence phenomena
The
fundamental breakdown to turbulence known as the \emph{K-mechanism} of
transition has been simulated under controlled conditions. The important
characteristic features of the K-mechanism, namely peak-valley formation,
significant spanwise modulation of the velocity, shear layer lift-up, and
appearance of spikes have been observed. A possible explanation of the
characteristic traits of this K-mechanism has been given using wave-resonance
theory. Due to resonance, initially large-amplitude harmonics of a plane 2D wave
are generated, and with increasing streamwise distance the harmonics of all the
higher spanwise wavenumbers appear gradually and amplify in a deterministic
manner. Up to the location of spike formation the amplitudes of the spanwise
modes increases, reaching their highest levels just after the appearance of the
spikes, while slightly dropping off further downstream.
kiran@mae.cornell.edu