Coarse grid projection methodology: A partial mesh refinement tool for incompressible flow simulations
NNoname manuscript No. (will be inserted by the editor)
Coarse grid projection methodology: A partial mesh refinementtool for incompressible flow simulations
A. Kashefi
Short Communication
Abstract
We discuss Coarse Grid Projection (CGP)methodology as a guide for partial mesh refinementof incompressible flow computations for the first time.Based on it, if for a given spatial resolution the numer-ical simulation diverges or the velocity outputs are notaccurate enough, instead of refining both the advection-diffusion and the Poisson grids, the CGP mesh refine-ment suggests to only refine the advection-diffusion gridand keep the Poisson grid resolution unchanged. Theapplication of the novel mesh refinement tool is shownin the cases of flow over a backward-facing step andflow past a cylinder. For the backward facing step flow,a three-level partial mesh refinement makes a previouslydiverging computation numerically stable. For the flowpast a cylinder, the error of the viscous lift force is re-duced from 31.501% to 7.191% (with reference to thestandard mesh refinement results) by the one-level par-tial mesh refinement technique.
Keywords
Coarse grid projection · Partial meshrefinement · Pressure-correction schemes · Flow over abackward facing step · Flow past a cylinder
In order to simulate incompressible flows using pres-sure correction schemes [1], the computational cost ona given coarse grid with N elements, C c , is approxi-mated by C c = C v + C p , (1) A. KashefiDepartment of Mechanical Engineering, Stanford University,Stanford, CA 94305, USAE-mail: kashefi@stanford.edu where C v and C p comprise the numerical cost of thenonlinear advection-diffusion equation and the linearpressure Poisson equation, respectively. Now, if the two-dimensional coarse grid is uniformly refined by l -level,the simulation using a high-resolution grid with M el-ements takes time C f , roughly determined as C f ≈ l C v + 4 l C p , (2)where 4 l is a factor for cost scaling of the advection-diffusion and the Poisson equations in a two-dimensional problem. According to the CGP technique[2–8], the advection-diffusion equation is executed onthe fine grid with M elements, while the pressure Pois-son equation is still solved on the coarse grid with N elements. Generally, we show the resolution of a CGPsimulation in the form of M : N , where M and N are defined as above. Hence, the computational cost ofCGP, C cgp , is estimated by C cgp ≈ l C v + C p + C m , (3)where C m is the mapping cost and is negligible in com-parison with the other two terms of Eq. (3). One mightsee Sect. 2.3 of Ref. [7] for further details.Let’s consider a condition that the standard numer-ical simulation diverged for the N : N case due to arelative high Reynolds number or too coarse a mesh.Or the results obtained with a N : N grid resolutionare not sufficiently resolved and a fluid field with moredetailed information is needed. The standard approachto resolving these common issues in pressure-correctionmethods [1] is to refine both the advection-diffusion andthe Poisson grids. In contrast with this approach, theCGP strategy suggests refining the advection-diffusiongrid, without changing the resolution of the Poissonmesh. To be more precise from a terminology point ofview, CGP does not propose a new mesh refinement a r X i v : . [ phy s i c s . c o m p - ph ] M a r A. Kashefi x/h -400 -200 0 200 400 m/s x/h -0.1 0.2 0.5 0.8 1.1 1.4 m/s (a)(b)
Fig. 1
Demonstration of partial mesh refinement application of the CGP method for the backward-facing step flow at Re =800, a comparison between axial velocity contours obtained using a Coarse scale computations (1600:1600), diverged, and b The CGP mesh refinement tool (102400:1600), converged. This figure is reproduced from Ref. [7]. method; however, it guides users to implement avail-able mesh refinement techniques for the grids associatedwith the nonlinear equations.From a mesh refinement application point of view,the cost increment factor of the computational CGPtool ( h cgp ) is approximated by h cgp = C cgp C c . (4)Similarly, this factor for a regular triangulation refine-ment ( h f ) is conjectured to be h f = C f C c . (5)Based on the above discussion, h f is greater than h cgp .This is mainly due to the factor of 4 l that multiples C p in Eq. (2). These results imply that mesh refine-ment using the CGP idea is more cost effective than thestandard technique. Note that we analyzed the compu-tational cost for finite-element discretizations. A simi-lar discussion is valid for finite volume/difference dis-cretizations [2, 3]. Here, we describe the concept by showing two simpleexamples. Both examples are taken from one of ourrecent published studies [7], but another interpretationof the numerical results of these examples is discussedhere.As the first example, consider the simulation of theflow over a backward facing step. Let’s assume that one oarse grid projection methodology: A partial mesh refinement tool for incompressible flow simulations 3
Table 1
Comparison of relative norm errors and h f /h cgp between the standard and CGP mesh refinement tools for thebackward-facing step flow at Re = 800. The norm errors are taken from Ref. [7]. * indicates that the simulation diverges after96 time steps. Resolution (cid:107) u (cid:107) L ∞ ( V ) %decrease in error (cid:107) u (cid:107) L ( V ) %decrease in error h f /h cgp time (s) C L f Fig. 2
Viscous lift coefficient obtained using the regularmesh refinement tool (215680:215680), the CGP mesh refine-ment tool (215680:53920), and the full coarse scale simulation(53920:53920). This figure is reproduced from Ref. [7]. is interested in the flow information at the Reynoldsnumber of Re = 800 (see Eq. (38) of Ref. [7] for thedefinition of the Reynolds number); however, due towall clock time or computational resource limitations,he is not able to run a simulation with the requiredpure fine 102400:102400 grid resolution. On the otherhand, because a coarse 1600:1600 resolution is not highenough, the simulation diverges after 96 time steps, asdepicted in Fig. 1a. The CGP framework with an inter-mediate resolution of 102400:1600 provides a convergedsolution as shown in Fig. 1b. The relative velocity er-ror norms with reference to the full fine simulation areof order 10E-5. Furthermore, the normalized reattach-ment length can be estimated around 14.0. Althoughthe error percentage of this estimation is 16.67% rela-tive to that obtained by the standard computations, itis captured 30 times faster. Note that these results areachieved by only refining the advection-diffusion equa- tion solver mesh. Table 1 lists relative norm errors ofthe velocity domain and h f /h cgp . For instance, refiningthe coarse mesh with the 6400:6400 spatial resolutionusing the CGP tool leads to a 1042.388% reduction inthe L norm error of the velocity field, while it is 28.613times cheaper than the regular mesh refinement tech-nique. Note that in the case of 1600:1600 spatial resolu-tion, because the simulation on the coarse grid diverges,there is no real number for C c ; however, if a virtual C c considered, h f /h cgp = 29 . C Lf ) at the Reynolds number of Re = 100 (see Eq.(39) of Ref. [7] for the definition of the Reynolds num-ber) for three different combinations of the advection-diffusion and the Poisson grid resolutions is depicted inFig. 2. Let’s assume an exact measurement of the liftcoefficient is needed for a specific engineering purpose.Using standard methods, this can be accomplished us-ing either 215680:215680 or 53920:53920 grid resolu-tions. An implementation with the finer grid producesa more precise answer. It could be a users incentive tolocally/globally refine the full coarse mesh. Obviously,this mesh refinement ends in an increase in CPU timefor the simulation. In this case, our numerical exper-iments show that the increase is equal to 339271.6 s(over 94 hr). As discussed in Sect. 3.3 of Ref. [7], hav-ing a coarse mesh only degrades the level of accuracyof the viscous lift not the pressure lift. In fact, insteadof refining the grids associated with both the nonlinearand linear equations, a mesh refinement of the nonlin-ear part is enough alone. Hence, in order to increase theprecision of the lift force, one can refine the advection-diffusion grid and keep the resolution of the Poissonmesh unchanged. In this case, the CGP grid refinementcost factor is h cgp = 3 . h f = 11 . A. Kashefi ing computation problems today. These problems aremerely used as examples to explain one of the featuresof the CGP algorithm. Practical applications of theCGP mechanism as a mesh refinement tool are expectedto be useful for three dimensional flow simulations onparallel machines.