Code Verification of ReFRESCO With a Statistically Periodic Manufactured Solution

Lu´is Ec¸aIST-ULMechanical Engineering DepartmentAv. Rovisco Pais 1, 1049-001 LisboaPortugalEmail: luis.eca@ist.utl.ptGuilherme VazMARINRD discretization errors (space and time) and the determina-tion of the observed order of (space and time) convergence.The availability of an exact solution allows the determina-tion of the numerical error and so the effects of iterative and dis-cretization errors can be addressed. The paper presents grid andtime refinement studies with different (iterative) convergence cri-teria and demonstrates that grid and time resolution are stronglyconnected when attempts are made to minimize the numericaluncertainty in the calculation of unsteady flows.The paper also addresses error estimation based on powerseries expansions in the calculation of unsteady (space and timedependent) flows. Simultaneous grid and time refinement is com-pared to grid refinement with fixed time step and time refinementwith fixed grid. The advantages and limitations of both optionsare discussed in the context of Code Verification (error evalua-tion) and Solution Verification (error estimation).INTRODUCTIONPapers presented at previous and present OMAE Confer-ences illustrate that the use of Computational Fluid Dynamics(CFD) tools in practical Engineering problems has become com-mon practice. But, considering the complexity of the flow prob-lems being handled, sooner or later the question about the trust-worthiness of CFD must arise. Is the CFD-result physically re-alistic or is it heavily biased by numerical errors of one kind oranother. That such question would arise has been anticipated andserious efforts have already been made to establish a methodol-ogy for estimating the confidence level of a numerical simula-tion. This is the realm of what is today generally denominatedby Verification and Validation (V&V) [1,2].The first step to check the trustworthiness of any CFD codeis Code Verification [1,2], i.e. the demonstration that the code isfree of errors. To achieve such goal, discretization errors, i.e. thedifference between the numerical and the exact solution must beevaluated. For the most applied mathematical model in CFD, i.e.the Reynolds-Averaged Navier-Stokes (RANS) equations, exactsolutions are as a rule not available. Fortunately, the Method ofthe Manufactured Solutions (MMS) [3, 4] offers an alternative.In the MMS, a continuum solution is first constructed, i.e. onespecifies all unknowns by mathematical functions. In general,this constructed solution will not satisfy the governing equationsbecause of the arbitrary nature of the choice. But by adding anappropriate source term, which removes any imbalance causedby the choice of the continuum solution, the governing equationsare forced to become a model for the constructed solution.In statistically steady flows, time-averaging is applied to theflow properties and to the conservation principles. As a conse-quence, time derivatives of the (mean) dependent variables van-ish and so (as for any steady flow) the RANS equations requirespatial discretization techniques only. In the open literature sev-eral examples of Code Verification exercises can be found forsteady (statistically or not) flow solvers, as for example [5,6,7,8].In all these examples, the correctness of the code is demonstratedwith numerical solutions obtained in a succession of systemati-cally refined grids. The main goal of the procedure is to demon-strate that the discretization error