On the Influence of the Iterative Error in the Numerical Uncertainty of Ship Viscous Flow Calculations

Transcription

1 26 th Symposium on Naval Hydrodynamics Rome, Italy, September 26 On the Influence of the Iterative Error in the Numerical Uncertainty of Ship Viscous Flow Calculations L. Eça (Instituto Superior Técnico, Portugal) M. Hoekstra (Maritime Research Institute, Netherlands) Abstract 1 Introduction This paper presents a study on the influence of the iterative error on the numerical uncertainty of the solution of the Reynolds-averaged Navier-Stokes equations. Two main topics are addressed: the estimation of the iterative error; and the influence of the iterative error on the estimation of the discretization error. Iterative error estimators based on the, and norms of the differences between iterations and on the normalized residuals are tested on three test cases: the 2-D turbulent flow over a hill, a 3-D flow over a finite plate and the flow around the KVLCC2M tanker at model scale Reynolds number. Two types of procedures are considered, one using the data of the last iteration performed and the other using an extrapolation based on a least squares fit to a geometric progression. In the latter case, the option of including the standard deviation of the fit in calculation of the error estimator is also tested. The results show that the most reliable estimates of the iterative error are obtained with the extrapolation technique including the effect of the standard deviation of the fit applied to the norm of the differences between successive solutions. To obtain realistic estimates of the iterative error one should avoid the use of the and norms and the use of the differences obtained in the last iteration. The results further suggest that the contributions of the iterative and discretization errors to the numerical uncertainty are not independent. Nevertheless, for the levels of grid refinement tested it is not necessary to converge a solution to machine accuracy to obtain a negligible contribution of the iterative error. It suffices to ensure that the convergence criterion guarantees an iterative error roughly 2 to 3 orders of magnitude below the discretization error. Model testing is still the most used and widely accepted approach in ship hydrodynamics investigations, but there is no doubt that the role of Computational Fluid Dynamics (CFD) in design problems is growing. In experimental fluid dynamics it is standard practice to indicate the uncertainty of a specific measurement. Therefore, it is hard to believe that CFD may establish itself as a reliable alternative and complement to model testing without indicating the numerical uncertainty of a given prediction. It is commonly accepted, [1], that the numerical uncertainty of a CFD prediction has three components: the round-off error, the iterative error and the discretization error. The round-off error is a consequence of the finite precision of the computers and its importance tends to increase with grid refinement. The iterative error is originated by the non-linearity of the mathematical equations solved by CFD. In principle, one should be able to reduce the iterative error to the level of the round-off error. However, this may not always be possible in complex turbulent flows. The discretization error is a consequence of the approximations made (finite-differences, finite-volume, finite-elements,...) to transform the partial differential equations of the continuum formulation into a system of algebraic equations. Unlike the other two error sources, the relative importance of the discretization error decreases with the grid refinement. In practical applications of complex turbulent flows, the discretization error is the most important of the three. Therefore, the main attention has been focused in recent years on the estimation of the discretization error, [1]. Methods based on grid refinement studies or single-grid error estimators have been tested in complex turbulent flows, like for example in the Workshop held in Lisbon in October 24, [2, 3]. Most of these exercises were performed with negligible contributions of the iterative error for the numer-

2 ical uncertainty, i.e. the solutions were converged to machine accuracy. The convergence criteria applied in practical CFD applications are usually much less demanding than requiring machine accuracy. The reasons for such practice are simple: the c.p.u. time to reach machine accuracy may be significantly higher than to satisfy an acceptable tolerance for practical purposes; the modelling error is often larger than the acceptable tolerance of the iterative error; the iterative convergence may stagnate at a level higher than machine accuracy. In such situations, the contribution of the iterative error to the numerical uncertainty of a given prediction may not be negligible. Furthermore, the iterative error may have an indirect effect on the estimation of the discretization error. Therefore, it is important to have reliable methods for the estimation of the iterative error and to investigate its influence on the estimation of the discretization error. The present paper addresses these two topics: estimation of the iterative error of a CFD prediction, and influence of the iterative error on the estimation of the discretization error. The estimation of the iterative error has some similarities with the estimation of the discretization error using grid refinement studies. However, there is an important difference: the discretization error based on grid refinement studies requires the calculation of the flow field on several grids; the iterative error, on the other hand, can be estimated for each single calculation. This inherent difference has the following consequences: In the estimation of the discretization error, the number of grids available (data points) is usually small. On the other hand, there are plenty of data available for the estimation of the iterative error of a given calculation. The discretization error is normally estimated at fixed locations and/or for integral quantities, which usually require post-processing of the data. On the other hand, iterative error estimators are best based on some norm for the complete flow field in each iteration cycle, to avoid the inconvenience of storing the flow field of all iterations. In this study, we consider the, and norms of two error estimators: the difference of the flow quantities between consecutive iterations; and the normalized residual of the discretized continuity and momentum equations. Two error estimations are performed for each type of norm and error estimator: the value of the norm in the iteration that satisfies the selected tolerance criteria; the sum of the infinite terms of a geometric progression with the first term (corresponding to the last iteration performed) and the ratio between consecutive terms obtained from a fit to the data of the available iterations. The iterative error estimators were evaluated in three test cases in which we were able to reduce the iterative error to machine accuracy: the 2-D turbulent flow around a hill, [4]; the 3-D flow over a finite flat plate; the flow around the KVLCC2M tanker at model scale Reynolds number, [5]. One and two-equation eddy-viscosity turbulence models, [7, 6, 8], are applied in the solution of the three test cases. The iterative error is evaluated for the three Cartesian velocity components and for the pressure coefficient. For the finite flat plate and the KVLCC2M tanker, we have also evaluated the iterative error of the viscous friction and pressure resistance coefficients. Obviously, the pressure resistance coefficient is computed only for the KVLCC2M tanker. The influence of the iterative error on the estimation of the discretization error is investigated for the same three test cases. The procedure proposed in [9] is applied to numerical solutions with different levels of iterative error. The resulting error bars are compared to evaluate the influence of the iterative error on the estimated discretization error. The purpose of this comparison is to check if the assumption that the two sources of error are independent is consistent with the uncertainty estimates obtained in the selected test cases. The present paper is organized in the following way: section 2 presents the procedures to estimate the contributions of the iterative error and discretization error to the numerical uncertainty. The three test cases selected and the flow solvers used are briefly described in section 3. The tests of the iterative error estimation are presented and discussed in section 4 and section 5 presents the estimation of the discretization error contribution to the numerical uncertainty for different levels of iterative error. The conclusions are summarized in section 6. 2 Estimation of the Numerical Uncertainty The numerical uncertainty of a CFD prediction is composed of three contributions: the round-off error, the iterative error and the discretization error. Usually the discretization error dominates, but the relative importance of the first two contributions tends to increase with grid refinement. Consequently, it is impossible to reduce the numerical uncertainty indefi-

3 nitely with the increase of the grid refinement. 2.1 Round-off Error The round-off error is related to the finite precision of the computers and to the condition number of the system of equations solved in a given application. In general, the condition number tends to increase with the size of the system and so the round-off error tends to increase with the grid refinement. The minimum level of the round-off error is dictated by the precision of the machine, which for a 32 bites machine in double precision is 15 digits. In general, this is enough to guarantee that the round-off error is negligible compared with the other sources of numerical error. 2.2 Iterative Error The iterative error is related to the non-linearity of the system of partial differential equations solved in CFD. There are several sources of non-linearity in the Reynolds averaged Navier-Stokes equations: The convective terms. The usual linearization procedures are Picard or Newton methods, see for example [1], which imply an iterative solution. The turbulence closure. For example, one and two-equation eddy-viscosity models have nonlinear convective terms and non-linear production and dissipation terms. Also, the turbulence model equations are often solved segregated from the continuity and momentum equations. Furthermore, the linear system of algebraic equations obtained from the discretization of the linearized partial differential equations is rarely solved with a direct method. Therefore, the flow solution includes an extra iterative cycle corresponding to the method applied in the solution of the linear systems of equations. In most flow solvers, no clear distinction is made between the various iterative cycles. Therefore, in the estimation of the iterative error it is important to point out the meaning of one iteration of the solution procedure. In principle, the iterative error may be decreased as far as the machine accuracy permits. However, in complex turbulent flows it is not guaranteed that one achieves that level of convergence. Furthermore, the c.p.u. time required to attain such level of iterative error may be significantly higher than to obtain an acceptable level of the iterative error. Therefore, it is unusual to converge industrial applications of CFD to machine accuracy, which brings up the need to develop reliable techniques for estimating the contribution of the iterative error to the numerical uncertainty. One possibility is to express the iterative error, e i, as a power series expansion e i = φ n i φ grid = j=1 α j n q, (1) where φi n stands for any integral or local flow quantity at iteration n, φ grid is the exact solution of the discretized non-linear system of equations on a given grid, α j are coefficients related to the level of the error and q is related to the rate of convergence 1. As in the estimation of the discretization error, [1], one could neglect all high-order terms to obtain e i φ n i φ go = αn q, (2) where φ go is the estimate of φ grid. But the use of equation (2) has several drawbacks: For local flow quantities, it requires the complete flow field for several iterations. In general, this is inconvenient because it implies the handling of huge amounts of information. For integral flow quantities, it requires the postprocessing of the data for each iteration of the solution procedure. For both local and integral quantities, it refers to the value of the flow quantities themselves rather than differences between consecutive iterations 2 on which convergence criteria are typically based. A more attractive way to estimate the iterative error is to use norms of the change in the solution from one iteration to the other. In this study, we have tested 1 A converging solution must have q <. 2 As we will discuss below, norms of the residual can also be interpreted as differences between iterations.

4 three norms 3 : ( φ) = MAX( φ i ) 1 i N P ( φ) = ( φ) = N P φ i i=1 N P N P i=1 ( φ i ) 2 N P (3) where N P stands for the total number of nodes of a given grid and φ for the local change of the flow quantity φ. Two options were used for φ: the variable change between consecutive iterations, φ d = φ n φ n 1, and the normalized residual of the discretized equations, φ r. In using these options, there are some important practical details. The difference in φ between iterations, φ d, is readily evaluated. However, it may be affected by the use of under-relaxation in the calculation procedure. If implicit under-relaxation schemes are applied, as for example local time-stepping, φ d will reflect its influence correctly. On the other hand, explicit under-relaxation must be handled carefully: φ d should be calculated before the under-relaxation is applied, otherwise, the values of φ d will become artificially small. The relation between the residual of the discretized equations and the flow quantities depends on the method adopted. Nevertheless, in general, the normalized residual, φ r, is equivalent to the differences in the solution of a Jacobi iteration for the system of equations of a given iteration. The term normalized means that the main diagonal of the system is scaled to one in order to obtain a right-hand side which represents a change in the dependent variable. Then φ r is also a measure of the differences between consecutive iterations. The values of, and obtained in any iteration n that satisfies the selected convergence criteria may be used as iterative error estimators. However, there is no guarantee that these values bound 4 the iterative error, specially when the rate of convergence 3 The designation of and norms is abusive, because their formal definition does not include N P in the denominator as a scaling. Our definitions make and equal to the mean value of φ and to the root mean square of φ, respectively. 4 One could argue that the goal of iterative error estimation is also 95% confidence and so we should say band. However, for this initial work we are trying to avoid philosophical discussions on the consequences of having 95% confidence on the iterative error and will try to check if the present error estimators bound the iterative error for the selected test cases. is small. Therefore we have adopted a common representation of the L( φ) norms as geometric progressions, L( φ) n = L( φ) no 1 q(n o n) (4) with a ratio, ρ, between consecutive evaluations of L( φ) n : ρ = L( φ) n = 1 q. (5) L( φ) n 1 L( φ) no is the value of the L norm at the last iteration n o and q is related to the convergence rate. It is obvious that a convergent solution requires ρ < 1, i.e. q <. Equation (4) may be re-written as log(l( φ) n ) = log(l( φ) no ) q(n o n), (6) which defines a linear relation with two unknowns, log(l( φ) no ) and q. It is not safe, however, to obtain q from two instances of equation (6), because there is no guarantee that the convergence is smooth 5. Moreover, there are plenty of data available from previous iterations. Therefore, the safest way is to solve equation (6) in the least squares root sense using the data of the last m + 1 iterations. Designating log(l( φ) i ) by L i, this leads to ( )( ) no (m + 1) il i L i i q f = and (m + 1) n o i=n o m n o i=n o m i 2 i=n o m ( no i i=n o m )( no no i=n o m i i=n o m ) (7) n o n o L i q f i i=n log(l( φ) no f ) = o m i=n o m + q m + 1 f m, (8) where L( φ) no f is the value obtained from the fit, which is not necessarily equal to L( φ) no. In some cases, the convergence may not be very smooth and so the standard deviation of the fit, D f, may be significant. D f = n o i=n o n ( ( )) 2 L i log(l( φ) no f ) + q f (n n o ) n 1 (9) For ρ < 1, the sum of all terms of the geometric progression (4) with n n o is given by L( φ) = L( φ) n o f 1 1 q f. (1) 5 We recall that several iterative cycles may be present in one iteration of the solution procedure..

5 The standard deviation of the fit, D f, may be added to log(l( φ) no f ) of equation (8) to obtain a more conservative estimate of L( φ) from equation (1). Furthermore, the slope q f may also be increased by 2D f /(n + 1), which is equivalent to adding and subtracting D f at the two ends of the fitted line in the most unfavourable way. In summary, we have tested as iterative error estimators L( φ) no and L( φ) of the, and norms using φ = φ d and φ = φ r Monitoring the Iterative Error For every grid of each test case, the solution was converged to machine accuracy. Therefore we can make an almost exact estimate of the iterative error for the solutions obtained with less demanding convergence criteria. We will designate the iterative error obtained from the difference with the solution converged to machine accuracy by e ie. The convergence criterion adopted will be designated by e t. A detailed description of the study performed for each selected flow variable for the three test cases is presented in [11]. In this paper, we will focus on two main quantities to monitor the iterative error: The maximum value of the ratio between the iterative error estimator, e i, and the iterative error obtained from the difference with the solution converged to machine accuracy, (e i /e ie ) max. The percentage of grid nodes where the iterative error estimator is larger than or equal to e ie, (e i e ie ), represented by the symbol F. The ratio (e i /e ie ) max gives an excellent indication of the quality of the iterative error estimator adopted. An optimal error estimator would have (e i /e ie ) max equal to one. If this value is smaller than one the error estimator does not bound the iterative error on all grid nodes. On the other hand, a value much larger than one indicates that the error estimation is too conservative, which may unjustly penalize the uncertainty estimation. If the error estimator fails to bound the iterative error, ((e i /e ie ) max < 1), 1 F quantifies the percentage of grid nodes where the estimate fails. As we have mentioned above, we consider two possibilities for the iterative error estimation. We will designate the error estimators based on the values obtained in the last iteration performed by e no i. The error estimates obtained from the sum of the infinite terms of a geometric progression are designated by e i. In order to simplify the notation, we will adopt the following designations: log((e no i /e ie ) max ) = R no ie and log((e i /e ie ) max )) = R ie 2.3 Discretization Error In the present investigation we have estimated the contribution of the discretization error to the numerical uncertainty, U d, using the procedure proposed in [9]. For the sake of completeness we describe it below. The estimation of the discretization error is based on a least squares version of the Grid Convergence Index method proposed by Roache, [1]. We use two error estimators: δ RE and M. The first is the error estimator obtained from Richardson extrapolation while M is the data range: φ i φ o = δ RE = αh p i, (11) ( ) M = max φ j φ i, 1 i, j n g, (12) where φ i is the numerical solution of any local or integral scalar quantity on a given grid (designated by the subscript i), φ o is the estimated exact solution, α is a constant, h i is a parameter which identifies the representative grid cell size, p is the observed order of accuracy and n g is the number of grids available. The data may suffer from scatter [12], which will only become apparent when solutions on more than three grids are available. Therefore φ o, α and p are obtained with a least squares fit of the data, minimizing the function: S(φ o,α, p) = ng i=1 ( φ i (φ o + αh p i ) ) 2, (13) Next, to identify the cases of oscillatory convergence or divergence, we also determine p using φi = φ i+1 φ i in (13); this fit includes only n g 1 differences. The apparent convergence condition is then decided as follows: 1. p > for φ Monotonic convergence. 2. p < for φ Monotonic divergence. 3. p < for φ Oscillatory divergence. 4. Otherwise Oscillatory convergence. Our procedure for the estimation of the numerical uncertainty, which has emerged from the experience obtained in several test cases and the suggestions and comments of the Workshop held in Lisbon in October 24, [2], and which is valid for a nominally secondorder accurate method, is then as follows:

6 1. The apparent convergence condition is identified, based on the observed order found, according to the definition given above. 2. If the convergence is monotonic: For.95 p < 2.5 U d φ = 1.25δ RE +U s. For < p <.95 U d φ =min(1.25δ RE +U s,1.25 M ). For p 2.5 U d φ =max(1.25δ RE +U s,1.25 M ) 3. If the convergence is not monotonic: U d φ = 3 M U s stands for the standard deviation of the fit performed to obtain p and δre is obtained with Richardson extrapolation using p equal to the theoretical value to obtain the error estimator, δre. 2.4 Numerical Uncertainty In this section we will assume that the round-off error is negligible compared with the discretization and iterative contributions to the numerical uncertainty, U. In all our previous exercises, see for example [9, 12], we have converged the solutions to machine accuracy, which means U U d. In this study, we consider situations where the iterative error is not negligible so that the numerical uncertainty will have contributions from the iterative and the discretization errors. Stern et al. [13] have proposed the following expression for the numerical uncertainty: U = (U d ) 2 + (U i ) 2, (14) which basically assumes that the two sources of uncertainty are independent. Since none of our selected test cases has an analytical solution, it is hard to check the validity of this proposal directly. But a good alternative is to estimate the contribution of the discretization uncertainty for different levels of the iterative error and to compare its values. If both contributions are really independent, the ratio between the discretization uncertainties should be one. 3 Flow Solver and Test Cases Three test cases have been selected to evaluate the procedures for the estimation of the iterative error and its influence on the discretization error: a 2-D turbulent flow over a hill, [4]; the flow along a finite flat plate and the flow around the KVLCC2M tanker. The first test case has been used in the Workshop dedicated to Verification of Calculations held in Lisbon in 24, [2], and it is part of the ERCOFTAC Classic Database, [14]. The flow along a finite flat plate has been addressed recently, [15], for the study of scaling effects based on CFD predictions. The KVLCC2M is one the test cases adopted for the Tokyo Workshop in Numerical Ship Hydrodynamics, [5]. All the calculations presented in this report were performed with the 2-D and 3-D finite-difference versions of PARNASSOS, [16], that solve the Reynoldsaveraged Navier-Stokes equations, written in contravariant form, coupled with the continuity equation in its original form (no artificial compressibility or pressure Poisson equation). Three eddy-viscosity turbulence models have been applied: the one-equation models of Spalart & Allmaras, [6], and Menter, [7], and the TNT version of the two-equation k ω model, [8]. One iteration of PARNASSOS involves the solution of two algebraic system of equations: The discretized turbulent quantities transport equations using the last available velocity and pressure field. The discretized continuity and momentum equations using the updated eddy-viscosity. Both systems are solved with a specified convergence criterion and a maximum number of preconditioned GMRES, [17], iterations, which, in general, are much less demanding than the convergence criteria applied for the flow field solution D Turbulent Flow over a hill The first test case is taken from the Ercoftac Classic Database, [14]. The computational domain is bounded by two solid walls and the inlet and outlet boundaries, which means that there are only two boundaries with exact boundary conditions. The Reynolds number based on the hill height, h = 28mm, and on the mean centreline velocity at the inlet, U o = 2.147m/s, is R n = 6. The inlet boundary is an x = constant line located at x = 3mm, i.e. x 1.7h and the outlet boundary is an x = constant line at x = 8mm (x 28.6h). The maximum distance between the two walls is 17mm, i.e. approximately 6h. Single-block structured grids have been adopted to generate a set of 7 geometrically similar grids, which

7 are a subset of the grid set B proposed for the Lisbon Workshop, [2]. One family of grid lines connects the inlet and outlet boundaries, whereas the other family of grid lines runs between the top and bottom walls. The coarsest and finest grids include and nodes, respectively. The maximum value of y + at the first grid node away from the wall is always below 1.35 for all the grids (below.6 for the finest grid). The calculations were performed with the following boundary conditions: No-slip at the top and bottom walls; the velocity and eddy-viscosity profiles are specified, [2], and the normal pressure derivative is set to zero at the inlet boundary; on the outlet boundary, the pressure is fixed and the remaining flow quantities are linearly extrapolated from the interior. 3.2 Flow over a finite flat plate The second test case is a three-dimensional turbulent flow over a flat plate of finite extension. The numerical calculation of this simple flow is important to address the problem of scaling effects, [15]. In the classical approach of determining the ship resistance from model test results, the viscous resistance is assumed to be proportional to the resistance of an equivalent flat plate. The computational domain is a parallelepiped of length 1.5L, where L stands for the length of the plate. The plate is.3l wide, but only half of it is inside the computation domain by adopting a symmetry plane. The cross-section of the computational domain is a square with side.25l. The inlet boundary is located.25l upstream of the leading edge of the plate and the outlet boundary.25l downstream of the trailing edge. Seven geometrically similar Cartesian grids have been used to discretize the computational domain. In all the grids, there is a grid line coincident with the leading and trailing edges and the tip of the plate. The grid node distribution in the longitudinal (x) direction follows a cosine distribution along the plate and onesided stretching functions, [18], are applied upstream and downstream of the plate. A semi-cosine distribution is applied in the transverse (z) direction between the symmetry plane and the tip of the plate. From the tip to the lateral boundary of the domain the grid node distribution is specified by a one-sided stretching function. In the normal (y) direction, one-sided stretching functions are applied to ensure the required y + for the correct application of the no-slip condition. In this case, the maximum value of y + at the first grid node away from the wall is below.8 for all the grids. Although we are aware of the shortcomings of such approach, no attempt was made to add a laminar-toturbulent transition model. Thus the turbulence model is supposed to take care of the transition simulation. The Reynolds number selected for this exercise is R n = (log(r n ) = 6.75). The boundary conditions are: Uniform flow at the inlet boundary; zero longitudinal pressure derivative applied at the outlet boundary, where streamwise diffusion is neglected; no-slip condition is applied on the plate surface; symmetry conditions are applied on the lateral boundary and at the grid nodes upstream, downstream and on the right of the flat plate; at the top boundary, we specify undisturbed flow velocity components in the longitudinal (x) and transverse (z) directions and undisturbed pressure. The coarsest and finest grids include and nodes, respectively. 3.3 Flow around the KVLCC2M tanker The final test case is the flow around the KVLCC2M tanker at model scale Reynolds number, Rn = This flow was one of the cases of the 25 CFD Workshop held in Tokyo, [5]. We have ignored the appearance of gravity waves, i.e. we have considered the double-model flow. Sets of 8 single-block grids have been generated for a computational domain that covers the complete starboard side of the ship. The outlet station is an x = constant plane located.25l downstream of the aft perpendicular, where L is the distance between perpendiculars. The external boundary is a circular cylinder with its axis on the symmetry plane of the ship at the water line and a radius of.18l. CO and HO HO(W) Grid N ξ N η N ζ N ξ N η N ζ G G G G G G G G Table 1: Number of grid nodes of the eight grids of the three grid sets for the flow around the KVLCC2M tanker. Three grid sets have been used in this study with different grid topologies. Two sets have an H-O topol-

8 ogy and the other a C-O topology. The two H-O sets differ in the grid node distribution in the streamwise direction, ξ. One set has a roughly equidistant grid node distribution, whereas the other one includes clustering of grid nodes at the bow and stern regions. The singular line of the wake extends from the keel line of the ship in all the grids. These grid sets have been used in previous studies performed for this flow, [9, 19]. Table 1 presents the number of grid nodes of the three grid sets in the streamwise, N ξ, normal, N η, and girthwise, N ζ, directions. CO designates the set with C-O topology, HO the H-O grids with equidistant grid nodes in the ξ direction and HO(W) the H-O grid set with grid refinement at the bow and stern. All the grids are generated with one-dimensional stretching functions applied in the normal direction, η, to guarantee the required value of y + for the application of the no-slip condition. In all the grids, the maximum value of y + at the first grid node away from the wall is always below 1. The boundary conditions applied in the calculation of the flow around the KVLCC2M tanker are: the velocity field at the inlet boundary is taken from a potential flow solution; at the outlet station, zero streamwise pressure derivative is imposed and diffusion in the streamwise direction is neglected; no-slip condition is applied on the ship surface. At the grid nodes of the bottom boundary on the symmetry plane of the ship, the three momentum equations are solved and the normal velocity is set equal to zero; at the external boundary, the tangential components of the velocity and the pressure are set equal to the potential flow solution; at the symmetry plane of the ship and on the free surface, symmetry conditions are applied directly to the tangential velocity components and to the pressure. The velocity normal to the symmetry plane is set equal to zero. 4 Iterative Error Estimation D Flow over a hill The calculations of the turbulent flow around the two-dimensional hill were performed with the eddyviscosity one-equation model of Spalart & Allmaras, [6]. In each of the seven grids tested, the calculations were started from scratch copying the inlet profiles to the complete flow field. In this test case, we have chosen to monitor the behaviour of the two Cartesian velocity components, U 1 and U 2 and the pressure coefficient, C p. All the quantities presented are dimensionless using the hill height and the mean centreline velocity as the reference values. The reference pressure to compute C p is the pressure at the outlet of the computational domain, i.e. C p = at the outlet. The convergence criterion is based on the maximum difference between consecutive iterations, ( φ d ). The calculation is considered to be converged when the following conditions are fulfilled: ( (U 1 ) d ) < e t ( (U 2 ) d ) < e t ( (C p ) d (15) ) <.1e t Three levels of e t have been selected: 1 3, 1 5 and 1 7. The reference solution was obtained with e t = Figure 1 presents the convergence of the, and norms of (U 1 ) d ( (U 1 ) r leads to similar results) for the finest and coarsest grids. The plots include also the least squares fits for the three levels of e t tested. The convergence behaviour of U 2 and C p is very similar to what is shown for U 1. Log[L( (U 1 ) d )] Log[L( (U 1 ) d )] =1-3 =1-3 =1-3 =1-5 =1-5 =1-5 =1-7 =1-7 = Iteration =1-3 =1-3 =1-3 =1-5 =1-5 =1-5 =1-7 =1-7 = Iteration Figure 1: Convergence history of U 1 for the flow around a two-dimensional hill and grids. Rn = 6.

9 There are several interesting features to observe: The graphical appearance of the plots of the finest and coarsest grid are similar. However, the convergence rate is drastically smaller in the finest grid. To attain machine accuracy, 181 iterations are performed for the coarsest grid, whereas the finest grid required about 234. It should be mentioned that the finest grid calculation is performed with more under-relaxation, but the use of a grid sequencing technique should enhance the performance of the calculation in the finest grid. Nevertheless, the reduction of the convergence rate with the grid refinement is consistently observed for the seven grids. As expected, we obtain > >, no matter what choice is made for φ. Qualitatively, the results obtained with (U 1 ) d and (U 1 ) r are similar. However, the iterative error estimations based on the two quantities are not equal. In both grids, the convergence rate of the three norms only stabilizes for small values of (U 1 ) d or (U 1 ) r. This means that the values of q obtained from the least squares root fit are clearly dependent on e t. Detailed results of the performance of the iterative error estimators with and without the standard deviations of the fits are presented in [11] for the selected flow variables in all the grids. As an example of the performance of the errors estimators tested, table 2 presents the monitoring parameters of the iterative error estimators of U 1 for the finest grid. n is the number of additional iterations performed to satisfy the convergence criteria. The results obtained in this flow exhibit the following trends: The iterative error estimators based on the value obtained in the last iteration performed, e no i, under-estimate the iterative error of the three flow quantities in all the grids. This is a systematic result observed in almost all the grids for every flow quantity, type of norm and type of difference between consecutive iterations. The most consistent results are always obtained with the norm. There are a lot of cases where the error estimate based on the norm bounds the iterative error of all the grid nodes, whereas the error estimates based on the and norms clearly fail to do so in a significant percentage of the grid nodes. The estimates based on φ d and φ r are fairly similar. However, in general, the most reliable ite t φ d n R no ie F o (%) R ie F (%) R no ie F o (%) R ie F (%) R no ie F o (%) R ie F (%) e t φ r n R no ie F o (%) R ie F (%) R no ie F o (%) R ie F (%) R no ie F o (%) R ie F (%) Table 2: Iterative error estimation of the U 1 velocity component grid. Flow around a twodimensional hill. Rn = 6. erative error estimates obtained are based on the difference between consecutive iterations, φ d. None of the estimates without the influence of the standard deviation of the fit bounds the iterative error for the three selected flow variables in all the grids. The best estimates are obtained with L( φ d ), which fail to bound the iterative error in 15 cases out of 63. Most of the failures occur for the lowest tolerance tested = 1 7. The estimates obtained with the influence of the standard deviation of the fit, D f, included in L( φ d ) (presented in [11]) show only one failure to bound the iterative error, R ie <. Furthermore, for this case F = 99.7%, which means that the estimate fails only in.3% of the grid nodes. On the other hand, the values of R ie obtained show that in some cases these iterative error esti-

10 mations may be too conservative. Nevertheless, it is safer to be too conservative in some cases than to under-predict the iterative error in a significant number of situations as in the estimates without D f. Figure 2 presents the iterative error of U 1 based on the solution converged to machine accuracy, eie, in the finest grid. With the data shown it is easy to assess the quality of the iterative error estimators based on the last iteration performed. For the three levels of et plotted, the values of L2 ( (U 1 )d ) of the last iteration performed are 1 4.3, and Log(eie(U1)): Log(eie(U1)): Log(eie(U1)): et=1-3 et=1-5 et=1-7 Figure 2: Estimated iterative error of U 1, eie, for different levels of the tolerance criteria, et. Flow over a two-dimensional hill grid. Rn = 6. For all levels of et, the maximum value of eie is two orders of magnitude larger than (U 1 )d of the last iteration performed. Furthermore, there is a significant region with eie > et that tends to increase for the less demanding tolerances. In total we have performed iterative error estimations for 21 cases: 7 grids with 3 levels of et. Table 3 presents the number of cases where F is equal to 1% for the three flow variables. The table includes Fo and F with and without D f. The aim of this table is to summarize the performance of the different methods tested if one wants to bound the iterative error. The data presented in table 3 confirm the main trends discussed above for this test case: Iterative error estimations based only on the values of the last iteration performed are not reliable. The L is best suited for the iterative error estimation. The L2 norm is clearly worse, but still a better choice than the L1 norm. U1 Fo F no D f F with D f U2 Cp U1 U2 Cp U1 U2 Cp L φ d L L L φ r L L Table 3: Number of cases where the iterative error estimators bound the iterative error (F=1%). 21 cases tested for each flow variable. Flow around a twodimensional hill. Rn = 6. The results based on φ d are globally the most consistent. The inclusion of the effect of the standard deviation of the fit, D f, in the extrapolation to an infinite number of iterations improves the reliability of the iterative error estimation. However, in some cases it may lead to a significant overestimation of the iterative error. 4.2 Flow over a finite flat plate The flow over a finite flat plate was computed with three eddy-viscosity turbulence models: the oneequation models of Spalart & Allmaras, [6], and Menter, [7], and the two-equation TNT k ω model, [8]. The iterative error is estimated for the three Cartesian velocity components, U 1, U 2 and U 3 and for the pressure coefficient, C p. The x axis is aligned with the flow direction, the y axis is normal to the plate and the z axis is aligned with the transverse direction. The reference velocity and length are the undisturbed velocity, U, and the plate length, L. As for the previous test case, the convergence criterion is based on L ( φ d ), but in this case it is determined only by the convergence of the pressure coefficient, L ( (C p )d ) < et. (16) Three levels of et have been tested: 1 3, 1 5 and 1 7. The reference solution was obtained with et = As an example of the convergence properties obtained in this test case, figure 3 presents the convergence of C p for the finest grid of the set with the

11 Log[L( (C p ) d )] Log[L( (C p ) d )] TNT k-ω Iteration Spalart & Allmaras =1-3 =1-3 =1-3 =1-5 =1-5 =1-5 =1-7 =1-7 =1-7 =1-3 =1-3 =1-3 =1-5 =1-5 =1-5 =1-7 =1-7 = Iteration Figure 3: Convergence history of C p for the flow over a finite flat plate in the finest grid of the set. Spalart & Allmaras and TNT k ω turbulence models. Rn = Spalart & Allmaras and TNT k ω turbulence models. The convergence properties of the three Cartesian velocity components are similar to those of the pressure coefficient. It should be mentioned that the 3-D PARNASSOS uses a grid sequencing technique in the solution procedure. Therefore, the initial approximation for step size 1 is much better than in the previous test case. In general, the trends observed in the convergence behaviour for the finite flat plate are similar to the ones discussed above for the previous test case. Nevertheless, there are some new aspects that we would like to emphasize: For a given grid, the convergence rate depends on the turbulence model selected. This means that estimates based on one turbulence model can not be extended to calculations performed with other turbulence models. In all the cases, the convergence rate is much higher in the initial than in the final phase of the iteration process. This means that the q values estimated with the largest values of e t will overpredict the convergence rate. Therefore, the iterative error may be under-predicted. The three norms respect the relation > >. In some cases, is more than one order of magnitude larger than and. Turb. φ d φ r Model MT F o SA 1 21 KW 2 32 F MT no SA D f KW F MT with SA D f KW Table 4: Number of cases where the iterative error estimators bound the iterative error (F=1%). 84 cases (4 flow variables) tested for each turbulence model. MT Menter one-equation model. SA Spalart & Allmaras one-equation model. KW TNT k ω two-equation model. 3-D flow over a finite flat plate. In general, the behaviour of the different iterative error estimators with e ie is similar to what was found in the first test case, [11]. In this paper, we will restrict ourselves to the data of table 4, which presents F o and F with and without D f for the three turbulence models. For each turbulence model, we have 7 grids with three levels of e t and 4 flow variables giving a total of 84 cases investigated. Although there is some influence of the turbulence model, the main trends are equivalent for the three turbulence models. Again, the most reliable results are obtained from the extrapolation to infinite iterations of the norm of the maximum difference between consecutive iterations, φ d, including the effect of the standard deviation of the fit, D f. 4.3 Flow around the KVLCC2M tanker The flow around the KVLCC2M tanker was computed with the one-equation model of Menter, [7], using the Spalart correction for the effects of streamwise vorticity, [2].

12 The iterative error is estimated for the three Cartesian velocity components, U 1, U 2 and U 3 and for the pressure coefficient, C p. The x axis is aligned with the undisturbed flow direction, the z axis is vertical positive pointing upwards and y completes a right-hand system. The origin of the coordinate system is located on the forward perpendicular at the ship symmetry plane on the free surface. All the variables presented have been made non-dimensional using the undisturbed flow velocity, U, and the distance between perpendiculars, L PP. As for the 3-D flat plate test case, the convergence criterion is determined only by the convergence of the pressure coefficient, ( (C p ) d ) < e t. (17) Three levels of e t have been tested: 1 3, 1 5 and 1 7. The reference solution was obtained with e t = For the flow around the KVLCC2M tanker at model scale Reynolds number we have performed calculation in three sets of 8 grids with different grid topologies: two H-O topologies using equidistant grid nodes in the streamwise direction (HO) and with clustering of grid nodes close to the bow and stern of the ship (HO(W)); a C-O topology with clustering of grid nodes at the bow (CO). As an example of the convergence properties in the calculation of the flow around the KVLCC2M tanker, figure 4 presents the convergence history of C p for the finest grids of the HO and CO sets. The convergence for the HO(W) set is similar to the one obtained in the CO set. The convergence rate is clearly dependent on the topology selected. In PARNASSOS, the reduction of the convergence rate is mainly related to the decrease of the streamwise grid line spacing. Nevertheless, it is possible to converge all the calculations to machine precision. Figure 5 presents e ie of U 1 in the finest grid of set HO(W) at the propeller plane, x =.98L, for the three highest levels of e t tested: 1 3, 1 5 and 1 7. The values of ( (U 1 ) d ) of the last iteration performed (in this case e t applies only to C p ) are 1 2.8, and 1 7, respectively. Logically, smallest values are obtained for the ( (U 1 ) d ) no : 1 3.8, and It must be emphasized that e t applies to the complete computational domain and we are only plotting the propeller plane. If the maximum changes occur at a different location, the maximum differences at the propeller plane can be substantially smaller than e t. The data plotted in figure 5 illustrate again the inadequacy of the iterative error estimators based only on the value of the last iteration performed. Log[L( (C p ) d )] Log[L( (C p ) d )] HO CO =1-3 =1-3 =1-3 =1-5 =1-5 =1-5 =1-7 =1-7 = Iteration =1-3 =1-3 =1-3 =1-5 =1-5 =1-5 =1-7 =1-7 = Iteration Figure 4: Convergence history of C p for the flow around the KVLCC2M tanker. Finest grids of the HO and CO grid sets. Rn = ( (U 1 ) d ) no underestimates the maximum values of e ie (U 1 ) by roughly one order of magnitude for the three levels of e t. Furthermore, there is a substantial part of the plotted region where ( (U 1 ) d ) no underestimates e ie (U 1 ). As for the finite flat plate, we present only the values of F o and F with and without D f for the three grid topologies tested in table 5. Detailed results of all the iterative error estimations are available in [11]. For each grid topology, we have 8 grids with three levels of e t and 4 flow variables giving a total of 96 cases investigated. In general, the data in table 5 show the same tendencies as the previous test cases. This is a clear indication that for the flow solver adopted in these tests the performance of the iterative error estimators is not significantly affected by the complexity of the computed flow. Nevertheless, there are some details in the data that we would like to highlight: There is a consistent comparison for all grid

13 z/l PP -.2 z/l PP z/l PP ( (U 1 ) d )=1-2.8 HO(W) y/l PP ( (U 1 ) d )=1-5.1 HO(W) y/l PP ( (U 1 ) d )=1-7 HO(W) y/l PP Log(e ie (U 1 )) Log(e ie (U 1 )) Log(e ie (U 1 )) Figure 5: Estimated iterative error of U 1, e ie, for different levels of the tolerance criteria, at the propeller plane, x =.98L. Flow around the KVLCC2M tanker grid of set HO(W). Rn = topologies of the three norms. The best performance is obtained with the norm and the worst with the norm. The three types of error estimation also show a consistent behaviour: the largest percentage of cases that bound the error is always obtained with the extrapolation to an infinite number of iterations using the least squares fit to a geometric progression including corrections based on the standard deviation of the fit; on the other hand, Turb. φ d φ r Model HO 7 1 F o HO(W) 4 15 CO 2 5 F HO no HO(W) D f CO F HO with HO(W) D f CO Table 5: Number of cases where the iterative error estimators bound the iterative error (F=1%). 96 cases (4 flow variables) tested for each grid topology. Flow around the KVLCC2M tanker. Rn = the lowest percentage of cases that bound the error is always obtained from the differences obtained in the last iteration performed, F o. In this case, the reliability of the iterative error estimators based on the extrapolation is a function of the grid topology selected. The HO and CO cases exhibit similar results, which are worse than the ones obtained for the HO(W) grid set. This is an interesting result, because, the HO(W) grids lead to the smallest convergence rate of the three sets. However, the HO(W) grids are the ones that present the least changes of the convergence rate of the three norms in the complete calculation of a given grid. Therefore, the hardest situations for the iterative error estimation 6 are not linked to level of the convergence rate, but to changes in its value along the iterative calculation. The flow variable and grid topology that present the largest number of cases where ( φ d ) with the effect of D f does not bound the iterative error is the U 2 velocity component in the CO set. It has F equal to 1% in 17 out of 24 cases, while the smallest value of F is 99.9%. It should be mentioned that the results contained in tables 3, 4 and 5 would present a lot more successful cases if the goal would be just F > 95%. However, the estimates based on the differences of the last iteration performed would remain almost unaffected. 6 Obviously, this does not mean that the estimated iterative error is independent of the convergence rate. This observation is related to its reliability, i.e. if it bounds the exact iterative error or not.