VERIFICATION AND VALIDATION CONSIDERATIONS REGARDING THE QUALIFICATION OF NUMERICAL SCHEMES FOR LES FOR DILUTION PROBLEMS

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1 VERIFICATION AND VALIDATION CONSIDERATIONS REGARDING THE QUALIFICATION OF NUMERICAL SCHEMES FOR LES FOR DILUTION PROBLEMS F. Ducros *, U. Beder, O. Con, T. Fortn, B. Fourner, P. Quéméré Commssarat à l Energe Atomque, DEN/DER/SSTH/LMDL, 17 rue des Martyrs, Grenoble cedex 9, FRANCE. Phone: +33 (0) , Fax: +33 (0) , E-Mal: frederc.ducros@cea.fr Abstract The Large Eddy Smulaton (LES) concept s, snce a long tme, consdered as a very promsng canddate for advanced thermalhydraulc modellng. However, lmted CPU resources and very long related elapsed tme have restrcted ts extensve use for ndustral purposes n general and for Nuclear Reactor Safety (NRS) ssues. The ncreased CPU resources have sgnfcantly mproved ts potental. The pcture today s that, even f some partcular turbulent flows (for examples hgh Reynolds number near wall domnated flows) stll compel to mx the LES approaches wth other approaches derved from Reynolds Average (RANS) models, perhaps the most compellng case for LES can be made for momentum, heat and mass transfer n free shear flows at hgh Reynolds numbers (from Pope (2004)). Under these basc condtons there are strong suggestons to make use of LES for dluton problems. Extensve verfcaton and valdaton attempts towards ths objectve as well as for ndustral applcatons (Beder et al. (2006 and 2007)) have already been performed for the Tro_U 1 code. Ths paper proposes a short overvew of what has been done, what can be related to the so-called Best Practce Gudelnes (BPG (2007)) n ths work and tends to answer specfc ssues of the LES valdaton for dluton ssues. 1 INTRODUCTION The modellng ssues comng along wth the use and the valdaton of LES for Boron dluton problems are numerous and cannot be totally treated through ths paper. We therefore have focused on the test and selecton of the convecton scheme. The results of a LES are known to be very senstve to ths parameter and the Boron concentraton calculated at the core nlet s manly the result of the Boron convecton and the related turbulent dffuson. The selecton of the numercal scheme s part of the numerous recommendatons comng along wth the use of LES for smulaton on NRS ssues (BPG (2007)). Low dffusve schemes are sad to be mandatory for LES of turbulent flows snce upwndng compromses accuracy (Garner et al. (1999) as well as Mttal and Mon (1997)) n the meanng that the numercal dsspaton may spurously nteract and eventually hde the molecular and sub-grd vscosty effects. Some works assocate ths property wth the search for quadratc quanttes conservaton at a dscrete level whch s an nterestng property as far as the knetc energy conservaton s concerned (see Ducros et al. (2000) for schemes related to structured meshes and Mahesh et al. (2004) for unstructured meshes). However, models based on numercal dsspaton comng along wth upwndng are very attractve to use snce they may behave well and are already avalable n all CFD codes: ths approach refers to Monotonc Integrated Large Eddy Smulaton (MILES, see: Fureby et al.(2005)). The use of such schemes may show robustness advantages but there seems to be a large consensus to prefer the prevous mentoned low dsspatve schemes when there are avalable. Recent works focus on the treatment of scalar advecton and suggest that the best compromse for LES s to make use of centred schemes for momentum transport and hgh order regularzng schemes for scalar transport (see: Chatelan et al. (2004)) for a proposal for structured meshes). Among the 1 1

2 reported reasons for ths choce are the prevous knetc energy conservaton for the momentum equaton and the search for a compromse between the requred postvty of the convecton scheme for the scalar, whch s mandatory to keep the scalar between physcal bounds and the fact that a too large numercal dffuson may, as for the momentum, spurously nteract wth the turbulent dffuson at the cut of level. We also have to bear n mnd that t s the scalar extremum (n fact the mnmum Boron concentraton) that s scrutnzed at the core nlet n the partcular case of the Boron dluton problem. A partcular attenton has therefore to be pad to the meanng of extrema n the LES framework and what are the man senstve parameters that nfluence drectly these values. In addton, usng CFD for ndustral purposes leads to the treatment of arbtrary complex geometres, what generally requres the use of unstructured meshes. Logcally, the verfcaton and valdaton (V&V) processes should be conducted n the sprt that the used mesh should exhbt the same features as the ones encountered n the eventual applcatons. Ths means that V & V tests should n partcular take nto account strong meshng ansotropy and cell sze varatons f such events are to be part of the ndustral model applcaton. After a bref recall of LES features, the paper gves a short overvew of what has been done for the valdaton of the LES scheme of the Tro_U code for free shear flows and summarses an applcaton of LES for dluton problems. A short analyss of what can be seen as conform to the BPG, as well as what can be ponted out as non-conform s proposed. Some related questons are then addressed and dscussed: What s the meanng of a fltered mnmum value n the LES framework? Is t possble to defne a BPG to approve a numercal scheme? How s the choce of a gven mesh and dscretsaton element determnant for the smulaton result? The paper fnally concludes wth some recommendatons whch are based on the posed questons. 2 SELECTED LES FEATURES WHICH ARE IMPORTANT FOR DILUTION MODELLING The frst queston, as also posed n Pope (2004), s: Is LES the rght approach? In accordance wth Pope, we are convnced that consderng the whole spectra of turbulent flows, t s now valuable to have a large range of approaches to study turbulent flows and therefore that there are models adapted to gven ssues. However, there s a larger and larger successful feld of LES applcatons, manly due to the ncreasng power of computers and the ongong work of scentfc communty. So let s frst stress some advantages brought up wth the LES underlyng concepts. To smplfy, one has to consder the result of any LES on a mesh of local sze h as the fltered counterpart u at a flter scale of the full resolved feld u. The basc decomposton comng along reads u = u + u where the fltered feld may be consdered as resultng from a convoluton product on u : Sagaut (2003). The range of moton scales of sze above the flter sze s explctly resolved. The nfluence of the unresolved scales on the resolved ones s modelled ether through the applcaton of SubGrd Scale (SGS) eddy vscosty models or through numercal treatments, or by a combnaton of both. Ths leads to potentally consder as dfferent from h. The topc of the mpact of the rato of / h on the soluton s rather complex and, n any case, not solved to date. Accordng to Pope (2004), the global pcture now suggests to dscard between: Pure physcal LES (the unresolved moton s explctly physcally modelled wth neglgble numercal errors), Physcal LES (the unresolved moton s explctly physcally modelled but where numercal errors are tolerated wthn the resolved scales) and Numercal LES (the descrpton of the resolved feld u depends on the numercal method, whch s typcally the case of MILES. sgs 2

3 Dealng wth ndustral flows nvolvng complex geometres yelds n usng most of the tme unstructured meshes and numercal schemes of lmted order. The treatment of these problems wth scheme of lmted numercal dffuson generally leads to consder that the resultng effectve flter s so that s close to h,.e. that the resultng LES s close to physcal LES n the Pope s classfcaton. Nevertheless, ndependent of the real classfcaton of the LES approaches, the LES concept leads to some very nterestng and common features we recall here: In the lmtng case when s gong to zero, u represents the complete feld u. Improved results are thus expected every tme the mesh s refned, whch tend towards the DNS result for moderate Reynolds number cases. Dependng of the selected modellng, the nfluence of the unresolved moton s treated through a mx of physcal models and the numercal treatment. These scales are close the homogeneous sotropc turbulence,.e. statstcally sotropc and unversal. One can consder the resultng modellng to be unversal, whch s a clear strength of LES approaches over RANS modellng. Snce the domnant phenomena for Boron dluton events are related to the mxng, where the turbulent transport s drectly affected by the large energy contanng scales, t s temptng to consder LES as a very promsng tool to deal wth Dluton ssues, provded a proper V & V process s followed. 3 BORON DILUTION TREATED BY LES CALCULATION In the frst CFD4NRS conference, a procedure to qualfy the CFD code Tro_U for predctve full scale reactor applcatons has been presented: Beder and Graffard (2006). In ths case, a plug of pure water s transported nto the lower plenum under natural convecton condtons. Calculatons under nomnal condtons are under way. The structures n the lower plenum are shown on Fg. 1 as well as an example of the Boron concentraton 80 seconds after the onset of natural convecton. Structures n the lower plenum The nstantaneous Boron concentraton feld Fg. 1: Structures of the lower plenum and Boron concentraton feld Such complex geometry can hardly been dscretzed on conform hexagonal structured meshes. Thus the numercal schemes of Tro_U are based on tetrahedral meshes. These elements allow avodng non-conformng meshes and large mesh ansotropes n physcally mportant regons. The correspondng meshng of the lower plenum s gven n Fg. 2, where, for nomnal mass flow condtons, an example of an nstantaneous velocty feld n the lower plenum s added (vectors and norm of the velocty n colors) 3

4 Meshng n the lower Plenum Velocty feld n the lower Plenum Fg. 2: Example of a tetrahedral meshng and a velocty feld n the mover plenum The qualfcaton of Tro_U for Boron dluton s the thrd step n a qualty assurance procedure whch ncludes the code verfcaton (test of the code on sngle effect test cases and analytcal solutons) and the code valdaton (ntegral test of the code on complex experments ncludng multple physcal phenomena). The qualfcaton s then a test strategy, whch guarantees the correct predcton of target quanttes for an engneerng problem. The flowchart for the qualfcaton of Tro_U for predctve full scale reactor calculatons s shown below on Fg. 3. The applcatve calculatons of the real reactor cases are lmted to predctvty due to the lack of publshed data of full scale experments. Adaptaton of the PWR model PWR Draft Model Refnement of the mesh Intal and boundary condtons Model hypotheses NO Is the modfed model consstent wth the PWR model? YES Valdaton calculatons UPTF model ROCOM model LACYDON model Modfcaton of the model Good agreement between all experments and calculatons? NO YES Qualfed PWR Model Refnement of the mesh Intal and boundary condtons Model hypotheses Fg. 3: Flowchart of a procedure to qualfy a code for predctve PWR calculatons When followng ths procedure, t s mportant to note, that the predctve calculaton s based on the same modelng hypothess as the valdaton calculatons. The valdaton on UPTF, ROCOM (Höhne et al. (2004)) and LACYDON experments (Beder and Graffard (2006)) s performed on ntegral effects 4

5 as the dstrbuton of scalars at the core nlet. In ths context, the meshng has been successvely refned n order to catch the large scale energy contanng scales. Then, the ntegral results are no more mesh dependent. When lookng to the flow feld n Fg. 2, these large scales are n the order of 0.2m to 1m, whch are well resolved by a meshng wth h of about 2cm. However, due to the lmtaton of the total mesh number, some mxng phenomena must be modeled, as e.g. the mxng n the wake of obstacles, where the scales may be of the order of some cm. The present resoluton ensures that the man structures of the flow responsble for mxng are captured by the mesh. Further detals on the numercal scheme behavor wll show that ths descrpton s close to the Pope s physcal LES. These results confrm Pope s ntuton (Pope (2004)) for compellng cases of momentum drven flows for mxng phenomena. In ths case, some Best Practce Gudelnes (BPG (2007)) recommendatons have been followed: Concernng the mesh desgn: the 3D meshes do not contan large growth factor between adjacent elements and contans a suffcent enough number of elements to solve the most knetc energy contanng eddes, whle descrbng the man flow gradents. A purely explct tme marchng method has been adopted, whch guaranty the correct tme ntegraton of movng eddes. However, we have to stress some dstances wth the BPG, as well as some open nduced questons whch wll be dscussed. What s the knowledge of extrema treatment n the LES framework and what are the most senstve parameters that are lnked wth extrema values? As prevously mentoned, t s sad, that LES should be used n conjuncton wth nondffusve hgh order (energy conservng, centred) schemes for space dscretzaton. The choce s more open for MILES modellng where more numercal dffuson s present. The queston s how to qualfy a gven numercal scheme wth respect to these features and should the good propertes for a scheme be the same for both, momentum and scalar transport? The underlyng sprt of the BPG leads readers to make use (f possble) of a pure, orthogonal hexahedral mesh (at least to avod tetrahedral meshes) for both numercal and CPU reasons. What are the consequences of the use of a pure tetrahedral grd? 4 WHAT MEANS A MAXIMUM (OR MINIMUM) VALUE IN THE LES FRAMEWORK AND WHAT IS THEIR RELATION WITH NRS ISSUES? sgs Let us consder the followng scale separaton φ = φ + φ as resultng from the applcaton of a convoluton product on the scalarφ. Let s take a scalar feld as φ( x) = sn( ωx) and apply a box flter between [ x Δ; x + Δ] to evaluate φ. The rato φ φ depends on sn( ω Δ) ωδ. Consderng a more complex sgnal contanng a range of scales up to the smallest scale η, the dependence of the rato φ φ may be then nvestgated n terms of η Δ. Under these condtons, whatever the propertes of the numercal scheme are (see below), there would not be any strct mesh convergence as long as the global approach lnks drectly the mesh and the flter szes (.e. Δ = Δ( h) ). Moreover, a bref analyss of φ φ suggests that the maxmum and mnmum values of φ are all the more reduced as the rato of η Δ s reduced. Applyng ths on reactor safety ssues means that flterng the contnuous feld φ wll result n a feld wth reduced maxma and mnma, what s aganst any ntutve conservatsm rules. Beyond ths ntutve analyss, the real mpact of ths potental based evaluaton of mnmum for NRS should be nvestgated. Indeed, snce local conservaton of Boron s ensured by most of the numercal schemes, local based mnmum values wll end n lttle too dffuse concentraton defects, whch s of no consequences f the sze of ths dffused spot s small enough. However, three types of mprovements seem possble: 5

6 A clearer dependency between the extrema of φ and the rato η should be nvestgated. If such a dependency may be modelled, n a form lke max φ φ fct( η ), then the local evaluaton of the Kolmogorov s scaleη wll allow to provde a more precse evaluaton of local extrema, A local refnement of the mesh n areas where the extrema are expected wll reduce ther under predcton. Such a refnement s partcularly easy to mplement when usng tetrahedral meshes. The modellng should be valdated not only aganst average quanttes, but also aganst extrema. Nevertheless, whatever the CFD code capabltes are to produce a locally fne mesh or to reconstruct an nterval for a gven extremum of φ, ths only makes sense f there s a strong confdence n the orgnal φ value. Ths partally results from transport capabltes of convecton scheme. Ths wll be addressed n the next paragraph. 5 HOW TO DEFINE A BPG APPROVED NUMERICAL SCHEME AND WHAT ARE THE LINKS WITH THE NRS ISSUE? An ntend of revew of exstng LES methods that pay attenton to dscrete energy conservaton for unstructured grd s provded n Mahesh, Constantnescu and Mon (2005), that presents a proposal for such a scheme n the frame of fnte volume (FV) methods. The method used n Tro_U reles on a fnte volume/fnte element (FV/FE) numercal scheme and offer possbltes to swtch between pure FE and mxed FE/FV formulatons. The varables are computed as a combnaton of base functons that are assgned to each element, dependng on the nodal values: the dscretzaton s based on a staggered postonng: P1-non-conformng for the velocty and P0/P1 for the pressure (Beder et al. (2007)). Ths dscretzaton method yelds a strong velocty/pressure couplng, whle ensurng global conservaton of the transported quanttes (momentum and scalar n our case). For momentum and scalar transport, the user has the choce between several convecton schemes of second order, whch go from purely centred, over the centred stablzed scheme presented n Kuzmn and Turek (2004) to standard 2 nd order upwnd scheme of MUSCL type. Detals for the numercal methods can be found n Heb (2003). 5.1 The momentum transportaton The knetc energy conservaton at a dscrete level can be ensured through the fnte element approach, provded that a centred scheme and ts antsymmetrc counterpart are used. Energy conservng scheme wthout SGSM EF_stab convecton scheme wth WALE SGSM Fg. 4: Knetc energy tme evoluton for a free decayng sotropc turbulence 6

7 Ths s shown n Fg. 4 for the case of the computaton of a freely decayng sotropc turbulence n a perodc cube of 32*32*32 hexahedra, where each hexahedron s cut nto 6 tetrahedral cells. The smulaton s performed wthout any SGS model (SGSM). Even f ths scheme provdes nterestng results for confguratons close to academc concerns, ts use for arbtrary complex geometres of applcatons n the nuclear feld encounters some lack of robustness and, as mentoned above, t s worth to add some stablzng procedures. Consderng the momentum equaton U r vr v r r v r v r P + U. U =. ( ν ( U + U ) T ) + β ( T T )g r 0, (1) t ρ the centred stablzed scheme used n Tro_U to dscretze the convecton term [ U v r. U v ] can be consdered as a blendng between a purely centred part and a mx of dffuson/ant dffuson terms: r r v [.( U antdffuson ] vr v v r r v v v [. U ] + ( U ).( U )) + SL( U )( U ) U α ). (2) centred dffuson The orgn of ths scheme s dscussed n the two papers of Küzmn and Turek (2004). Our formulaton (called EF_stab) offers the possblty of combnng a centred scheme wth the stablzng effects of upwnd decentrng, dependng on the parameters α et SL: If α s zero, a pure centred scheme s actvated. The slope lmter SL( U v ) automatcally deactvates the ant-dffusve term what yelds an upwnd scheme n regons of hgh gradents. In Tro_U, usually, the SUPERBEE lmter s used. The parameter α further weghts the reducton of the decentrng part of the scheme. Values of α = 1.0; 0.2 and 0.1 have been tested extensvely. In order to evaluate the qualty of the scheme, let us consder frst the knetc energy tme evoluton for a free decayng sotropc turbulence n the perodc cube of 32*32*32 cells hexahedral cells, each cell cut nto 6 tetrahedrons. Fg. 4 shows the temporal knetc energy decrease for several values of α n conjuncton wth a standard SGSM (WALE model from Ncoud and Ducros (1999)), together wth the t -1.4 reference slope. As expected, the UPWIND scheme shows a partcular trend to dsspate the knetc energy more than the others, whereas no clear dscrepances can be exhbted for the others. Smlar pcture (not shown) of a smulaton wth no SGS model wll more or less leads to the same conclusons, suggestng that, once one has njected some non lnear stablzng process n the scheme, the dffuson s enough to extract knetc energy and acts as a SGS model, at least for ths global measure of knetc energy. EF_stab convecton scheme wthout SGSM EF_stab convecton scheme wth WALE SGSM Fg. 5: Knetc energy spectra after sx turn over cycles for a free decayng sotropc turbulence 7

8 For a more profound test of the scheme, Fg. 5 shows the knetc energy spectra at about 6 turns over tmes for smulatons wthout (left) and wth WALE SGSM (rght), together wth the reference k -5/3 slope. For low values of α the slope at small scales s controlled by the SGS modellng, the whole modellng assurng the turbulent spectrum to follow a more or less constant slope up to the cut-off scale. Ths suggests consderng / h to be of the order of one. Hence, the whole modellng corresponds to physcal LES n the Pope s classfcaton, whch s close to an optmal compromse. 5.2 The scalar transportaton The frst measure of the qualty of a scheme for scalar transport concerns ts capacty to fulfl the requrement of physcal bounds respect, a property related to the postvty of the scheme. A former publcaton (Chatelan et al. (2004)) shows that centred schemes where not able to keep the scalar nto the physcal bounds, even n the case of completely developed turbulence, where sharp gradents are not attended. The paper therefore suggests that a compromse s to be searched for a regularzng scheme that s not to dffusve for the scalar transport. Defnng such a compromse for unstructured data s not straghtforward, and t can be shown that some reputed TVD schemes exhbt not strctly TVD behavour when dealng wth arbtrary flows on arbtrary mesh. The proposed test conssts n the convecton of a scalar spot (values between 0 and 1) n the man dagonal of a cube. A perspectve vew of the meshng as well as a typcal result of the scalar concentraton s gven n Fg. 6 for statonary condtons. Fg. 6: Transport of a scalar along a cube dagonal. Three schemes avalable n Tro_U are compared n Fg. 7, where the concentraton felds n the dagonal cut plane are shown. The fst order UPWIND scheme s postve but very dffusve. Our mplementaton of the MUSCL scheme wth the lmters van Leer and mnmod (Kuzmn and Turek (2004)) shows lmted but reproductve out of bounds values. It should be stressed that other mplementatons of the same lmters may modfy the compromse between respects of the physcal bounds and dffusve features and lead to a strctly postve but more dffusve MUSCL scheme, whch confrms a strong dependency on the mplemented lmters. The centred stablzed scheme EF_stab wth α=1 respects the bounds and gves not too dffusve physcally correct results. 8

9 UPWIND scheme (mn, max) = (0,1) MUSCL scheme wth van Leer slope lmter (mn,max) = (-0,034, 1,095) MUSCL scheme wth mnmod slope lmter (mn,max) = (-0,0076, 1,007) Ef_stab wth α=1 (mn,max)= (-2,e-18 ; 1) Fg. 7: Transport of a scalar along a cube dagonal (dagonal cut plane) These transport features have also been tested n the case of the scalar (temperature) transport n a freely decayng sotropc turbulence where the ntal condtons consst n a scalar spot that exhbts contact dscontnutes. The temporal evoluton of the extrema (maxmum and mnmum temperature) s show on Fg. 8. Overshoots for the MUSCL scheme and hgh dsspaton for the UPWIND scheme are detected. Fg. 8: Tme evoluton of the scalar extrema durng a freely decayng sotropc turbulence 9

10 From a NRS pont of vew, t s clear that the less dffusve scheme that respects physcal constrants has to be chosen snce the applcaton of too dffusve schemes wll reduce the occurrence and ntensty of extrema. For the Boron dluton problem ths results n an under predcton of the probablty of low concentratons concurrency as well as n a poor evaluaton of the area wth low concentraton values. As a concluson and regardng the state of the art for LES modellng, our best compromse leads to make use of stablzed scheme for both momentum and scalar convecton. For the momentum equaton, the stablzng procedure s tuned so that the global modellng s of physcal LES accordng to Pope s classfcaton, whereas the postvty of the scheme s strctly appled for the scalar transport. 6 HOW TO DECIDE IF TETRAHEDRONS OR HEXAHEDRONS ARE THE BEST CHOICE FOR INDUSTRIAL APPLICATIONS? The role whch plays the local mesh sze and the convecton scheme on the evaluaton of extrema of the feld φ has been dscussed. Both are related to the used meshng. Generally, Best Practce Gudelnes (BPG (2007)) recommend the use (f possble) of a pure, orthogonal hexahedral mesh and at least to avod tetrahedral meshes. Is ths recommendaton always a good choce? Snce only tetrahedral meshes have been tested for LES n the Boron dluton problem wth Tro_U, there s not enough nformaton to really dscard between hexahedral or tetrahedral elements. The flow n the lower plenum of a vessel does not exhbt any man preferental drectons and therefore the valdaton should take ths fact nto account. We further stress three nterestng features comng along wth tetrahedral scheme: Tetrahedral meshes are much easer to generate. In NRS analyss, often ¾ of the requred man power s related to mesh generaton. A smplfcaton n ths feld would dramatcally reduce the costs for CFD calculatons. Local mesh refnement technques can be easly consdered n tetrahedral grd. Such a local mesh refnement s necessary for a correct determnaton of extrema. Recent numercal developments as presented n ths paper show that good convecton schemes are also avalable for tetrahedral meshes. 7 RECOMMENDATIONS AND CONCLUSIONS A LES classfcaton n the sprt of Pope (2004) has been recalled. Ths classfcaton suggests that the optmal LES methodology reles on physcal LES whch supposes a flter to local mesh element sze of about 1. Before applyng such an LES on an ndustral bass, we propose smple tests to check the behavour of the used LES modellng and suggest for these applcatons the acceptance of lttle amount of numercal stablzaton. The proposed test smulates the freely decayng sotropc turbulence n a perodc box For the scalar transport we frst have underlned the role played by the LES flter of the extrema of a LES feld. We then have focused on the scalar transport scheme features to underlne the dffculty to get a real 3D TVD scheme for arbtrary tetrahedral mesh wth lttle dsspatve features. We have proposed such a scheme and show that the modellng results from a compromse between numercal dffuson and respects of physcal bounds. We have proposed to measure ths compromse wth tests aganst an sotropc freely decayng turbulence ncludng passve scalar transport, and aganst more standard scalar spot transport cases n non trval mesh drectons. As stressed n the BPG (2007), mesh convergence can not be followed n general for LES. In practce ths means that we are satsfed wth the convergence to DNS that comes along wth physcal LES as defne here. Even f ths can be seen as an acceptable LES feature wth regards to resolved varables when lookng at ther average values (for varables as well as for hgher moments), t should be looked wth care to the convergence to scalar extrema. The use of tetrahedral element s presented here as a clear advantage n terms of meshng capablty. The numercal scheme s of FE type, whch does not lead to compel wth orthogonalty constrants concernng grd angles. Under these condtons and consderng the present qualfcaton, usng ths 10

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