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docs:classes:dns [2009/02/16 19:17]
gibson
docs:classes:dns [2010/02/02 07:55] (current)
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   for (Real t=T0; t<=T1; t += N*dt) {   for (Real t=T0; t<=T1; t += N*dt) {
-     ​dns.advance(u,​p,​N);​+     ​dns.advance(u,​p,​N); ​                // advance u,p forward N steps of length dt
      ...      ...
   }   }
Line 36: Line 36:
  
 ===== DNSFlags ===== ===== DNSFlags =====
 +
 +The differents DNS flags are :
 +  * baseflow
 +  * timestepping
 +  * initstepping
 +  * nonlinearity
 +  * dealiasing
 +  * taucorrection
 +  * constraint
 +  * verbosity
 +
 +
  
 ===== Time-stepping algorithms ===== ===== Time-stepping algorithms =====
 +
 +The DNS class implements seven different time-stepping algorithms. (The default is SBDF3.)
 +
 +  * **CNFE1 or SBDF1**: 1st-order Crank-Nicolson,​ Foward-Euler or 1st-order Semi-implicit Back-
 +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.
 +
 +  * **CNAB2** 2nd-order Crank-Nicolson,​ Adams-Bashforth. A popular algorithm, but higher-frequency
 +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.
 +
 +  * **CNRK2**: a three-substep,​ 2nd-order semi-implicit Crank-Nicolson,​ Runge-Kutta algorithm, devel-
 +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.
 +
 +  * **SMRK2**: a three-substep,​ 2nd-order semi-implicit Runge-Kutta developed by Spalart, Moser, and
 +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.
 +
 +  *** SBDF2, SBDF3, SBDF4**: 2nd, 3rd, and 4th-order Semi-implicit Backward Differentiation
 +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 ===== ===== Nonlinearity =====
  
-===== Base flow =====+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.
  
-===== Mean constraint =====+  * **Rotational:​** Fast but generates high-frequency errors unless dealiased 
 +  * **SkewSymmetric:​** Comparatively expensive to compute compared to Rotational 
 +  * **Convective** 
 +  * **Divergence** 
 +  * **Alternating** convection/​divergence an alternating time steps. A cheap approximation to SkewSymmetric,​ which is an average of the convective and divergence forms. Not yet analyzed how the alternating nonlinearity method interacts with multistepping algorithms.
  
 +  * **Linearized** about the base flow.
  
  
  
 +===== Base flow =====
  
 +  * **Zeros**
 +  * **PlaneCouette** : plane Couette mean velocity profile <​latex>​ y </​latex>​
 +  * **Parabolic** : plane Poiseuille mean velocity profile <​latex>​ 1-y^2 </​latex>​
 +
 +
 +===== 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.
  
  
docs/classes/dns.txt · Last modified: 2010/02/02 07:55 (external edit)