Table of Contents

DNS and DNSFlags

The DNS1) class advances velocity fields under the Navier-Stokes equations, with a variety of times-stepping algorithms, boundary conditions, and external constraints. There are quite a few options to the DNS class, so many that there is a special DNSFlags class to group them all together.

DNSFLags and DNS work together like this. You construct a DNSFlags object and set its member variables to values that define the flow you want to integrate. Then you construct a FlowField, which defines the grid and cell sizes. Lastly you construct a DNS object combining the DNSFlags, the velocity field, the kinematic viscosity, and the time step. In C++ code it looks like this 

  DNSFlags flags;
  flags.baseflow     = PlaneCouette;
  flags.timestepping = SBDF3;            // 3rd order semi-implicit backwards differentiation
  flags.initstepping = SMRK2;            // 2nd order spalart-moser runge-kutta
  flags.nonlinearity = Rotational;
  flags.dealiasing   = DealiasXZ;
  flags.constraint   = PressureGradient; // enforce constant pressure gradient
  flags.dPdx = 0.0;                      // hold mean pressure gradient at zero
 
  FlowField u(Nx,Ny,Nz,3,Lx,Lz,a,b);     // velocity
  FlowField p(Nx,Ny,Nz,1,Lx,Lz,a,b);     // pressure
 
  DNS dns(u, nu, dt, flags);
 
  for (Real t=T0; t<=T1; t += N*dt) {
     dns.advance(u,p,N);                 // advance u,p forward N steps of length dt
     ...
  }

Sections to be written…

DNSFlags

The differents DNS flags are :

Time-stepping algorithms

The DNS class implements seven different time-stepping algorithms. (The default is SBDF3.)

ward Differentiation Formula –two names for the same algorithm. This algorithms is extremely simple and needs no initialization need, but its 1st-order error scaling makes it practically worth- less, except for initializing other algorithms.

modes are poorly damped. Requires one initialization step. Zang warns against us- ing CNAB2 in combination with Rotational nonlinearity unless the high-frequency modes are dealiased. CNAB2 enforces zero-divergence at successive timesteps and momentum equa- tions halfway between successive time steps, which can lead to slowly decaying period-2dt os- cillation in the pressure field, unless pressure and velocity are initialized accurately.

oped by Zang and Hussaini and but implemented in Channelflow from the Peyret’s exposition . According to Peyret, Zang and Hussaini observed 3rd-order scaling for this algorithm applied to low-viscosity flows, even though it is theoretically 2nd-order. Numerical tests in Channelflow show 2nd-order scaling for velocity fields at Re = 103 − 104 , and 1st-order scaling for pressure, due to a phase error in the pressure field. CNRK2 requires no initialization.

Rogers. Identical characteristics as CNRK2, including observed 2nd-order scaling consistent with theory, contrary to authors’ claim of 3rd-order scaling, and 1st-order phase error in pressure. Requires no initialization.

Formulae, requiring 1,2, and 3 initialization steps. I have found the SBDF schemes to be the best-behaved of the lot. When solving un+1 and pn+1 , SBDF schemes enforce divergence and momentum equations at tn+1 . This strongly implicit formulation poduces strong damping for high-frequency modes and results in pressure field as accurate as the velocity field. SBDF3 is par- ticularly good: it has the strongest asympotitc decay of all 3rd-order implicit-explicit linear multi- step schemes. For these reasons, SBDF3 is the default value of flags.timestepping. Peyret terms these algorithms AB/BDEk (kth-order Adams-Bashforth Backward-Differentiation).

To summerize : CNFE1, CNAB2, CNKR2, SMRK2, SBDF1, SBDF2, SBDF3, SBDF4

Nonlinearity

The nonlinear term in the Navier-Stokes calculation can be computed in a number of forms that are equivalent in continuous mathematics but slightly different when computed with spectral expansions and collocation. The default is SkewSymmetric.

Base flow

Mean constraint

Periodic channel flows satisfy the Navier-Stokes equations with either the bulk velocity or the spatial-mean pressure gradient set as an external constraint. This flag sets which constraint is to be enforced. DNS’s default behavior determines the spatial-mean pressure gradient or bulk velocity from the fluctuation’s initial condition u and matches this as a fixed constraint at each time step. DNS can match time-varying constraints as well.

1)
Direct Numerical Simulation