NUMERICAL INVESTIGATIONS WITH MODEL COMPARISONS ON FLUID FLOW OVER BACKWARD FACING SHARP EDGE STEP

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1 NUMERICAL INVESTIGATIONS WITH MODEL COMPARISONS ON FLUID FLOW OVER BACKWARD FACING SHARP EDGE STEP Dr. Nrmal Kumar Kund Assocate Professor, Department of Producton Engneerng Veer Surendra Sa Unversty of Technology, Burla , Inda Abstract A 2D numercal model s developed to examne supersonc flud flow over backward facng sharp edge step utlzng both RANS and hybrd RANS-LES turbulent models. Both the models also nclude added vtal factors specfcally producton, dffuson and destructon terms apart from the usual aspects correspondng to the current physcal problem. The smulatons are carred out wth the specfed models by ntroducng the nflow free stream Mach number of 2.5 correspondng to free stream pressure and velocty of N/m 2 and m/s 2, respectvely. Besdes, the grd dependence test s performed to the get optmal grd, whch s found to be of ( ), n the current stuaton. The smulaton results are matched wth the related expermental results exstng n the lterature. Despte the fact that both the model results are agreeable, nevertheless, the hybrd RANS-LES model brngs relatvely better and consstent results over the RANS model all over the whole flow regme. In addton, for better smulaton accuracy, solver convergence and resoluton near the wall vcnty, the non-dmensonal sublayer-scaled dstance y + of 1 and 11 (assocated wth the hybrd RANS-LES) are consdered n the present nvestgatons. Furthermore, the observed uneven flow behavours relatng to the pressure recovery s owng to the sudden expanson flow over the sharp edge step. Keywords Backward facng, Sharp edge step, RANS, Hybrd RANS-LES, Grd ndependence test, Pressure recovery. I. INTRODUCTION Flud flow over backward-facng step s one of the central contexts and has got partcular attenton as a result of not ust smplcty but for ample technologcal uses. In appled aerodynamcs, t s also used to study many complcated structures, ncludng separaton and reattachment. In the feld of research of hgh Mach number flow, the backward facng step s always consdered as a complex confguraton for gnton n a scramet, where the recrculaton vcnty has a sgnfcant role n stablzng the frng of the engne. Steps on the surfaces of hypersonc or supersonc arcrafts form the flow regme more complex and henceforward vtal nvestgatons are desperately necessary for mprovng the dynamc desgn of arcrafts. Smth [1] executed expermental examnatons on the flow feld and heat transfer downstream of a rearward facng step n supersonc flow. Launder and Sharma [2] used the energy dsspaton model of turbulence to analyse the flow feld around a spnnng dsc. Armaly et al. [3] conducted both expermental and theoretcal studes on backward facng step flow. Spalart and Allmaras [4] ntroduced a one-equaton turbulence model for assessng aerodynamc flows. Anderson and Wendt [5] reported llustrous and comprehensve descrptons of computatonal flud dynamcs. Neumann and Wengle [6] used both DNS and LES for examnng passvely controlled turbulent flow of backward-facng step. Hamed et al. [7] performed the numercal smulatons of fludc control for transonc cavty flows. Chen et al. [8] studed expermentally on fne structures of supersonc lamnar as well as turbulent flow over a backward-facng step by usng Nano-based Planar Laser Scatterng DOI: /IJRTER WYRLO 178

2 (NPLS). Lu et al. [9] nvestgated numercally on the nfluences of nflow Mach number and step heght on supersonc flows over a backward-facng step. Terekhov et al. [10] executed the expermental revsons on the separated flow behavour behnd a backward-facng step over and above the passve dsturbance. II. OBJECTIVES From the detaled examnatons, to the best of author acquantance, t s found that there s not a sngle full numercal report on flow over a backward facng sharp edge step expendng hybrd RANS-LES method. Wth ths vewpont, the present nvestgaton demonstrates the numercal studes on flow behavours over a backward facng sharp edge step exhaustng hybrd RANS-LES technque. Addtonally, the numercal model also comprses addtonal essental features namely producton, dffuson and destructon terms besdes the normal ssues relatng to the current research. Besdes, the stated model also contans both compressblty and eddy vscous effects. The model s thoroughly demonstrated for the metculous numercal studes on flud flow characterstcs relatng to flow over a backward facng sharp edge step by ncorporatng the nflow free stream velocty n conuncton wth the correspondng Mach number as the strategc model parameters. Eventually, the present case of backward facng sharp edge step for both RANS and hybrd RANS-LES predctons of fully supersonc turbulent flow are compared wth expermental data avalable n lterature. Ultmately, the smulaton predctons wth regard to the stated key model parameters are also along the expected lnes and are n very good agreement wth the assocated expermental results. III. DESCRIPTION OF PHYSICAL PROBLEM Backward facng sharp edge step devsng wde varety of uses n appled aerodynamcs s studed n the present research. The geometrc confguraton along wth ntal and boundary condtons are referred from the expermental examnaton report of Smth [1] Geometrc model Fgure 1 eptomzes the system confguraton for analysng the backward facng sharp edge step flow over sharp edge geometry separatng at a step heght H = m, upstream dstance from nlet to step Lu = m and downstream dstance from sharp edge step to outlet Ld = m. The dstance from downstream to upper boundary layer Z = m, spanwse dstance L= m and wdth B = m. The separaton and reattachment ponts are symbolsed by S and R respectvely and are expected to be pragmatc after executon of numercal smulaton. Fgure 1. Flow specfcaton of backward facng sharp edge All Rghts Reserved 179

3 3.2. Intal and boundary condtons The nflow free stream velocty Un = m/s, for whch the dentfed statc free stream pressure pn = N/m 2 corresponds to the Mach number Ma = 2.5. At the left sde ahead of the step, the ntal temperature s mantaned at K. The ntal condtons whch are set on the upstream are very much useful throughout the smulaton along the spanwse drecton, for accomplshment of the flow features beyond the step. For the turbulence, both RANS standard k-ε two-equaton model and Spalart-Allmaras one-equaton hybrd RANS-LES (otherwse termed as Detached Eddy Smulaton, DES) model are consdered. The boundary condtons for the geometry llustrated n fgure 2 are as follows: Pressure p = kpa, everywhere else for pressure n case of both RANS and hybrd RANS-LES models. Temperature Tn = K, everywhere else for temperature for both RANS and hybrd RANS-LES models. Velocty Un = m/s at the nlet, no-slp wall at the lower boundary, slp wall at the upper boundary and zero velocty gradent at the outlet are set for both the models. Fgure 2. Backward facng sharp edge step boundary representaton IV. MATHEMATICAL FORMULATION The most generalzed governng transport equatons of mass, momentum and energy stated n the conservatve form of Naver-Stokes equaton for compressble flow accompanyng the effects of turbulence are as mentoned below. ( u ) Contnuty: 0 (1) t x u Momentum: t E Energy: t Where, Total energy, x u u p p T T ( uu x ) p x x (2S ) T u E p k kt Stu Sh u p T 2 p v E ek h 2 x x t (2) 2 (3) (4) All Rghts Reserved 180

4 The Reynolds stress term s modeled n terms of the eddy vscosty and s expressed as: t 2 t( SSnn /3) 2k / 3 (6) The eddy vscosty s defned as a functon of the turbulent knetc energy k, and the turbulent 2 dsspaton rate ε, and s expressed as: t c fk / (7) In addton, all the model terms/symbols/coeffcents/functons have ther usual meanngs and values RANS Turbulence Modellng The standard k-ε model s well-known and s used extensvely for two-equaton eddy vscosty model. Transport equatons are nterpreted by two scalar propertes of turbulence.e., the transport equaton k-equaton s a model for the turbulent knetc energy and the ε-equaton s a model for the dsspaton rate of turbulent knetc energy. The turbulent transport equatons for the k-ε model are defned as follows: k t k Turbulent knetc energy: u k t S k t x (8) k x 2 Energy dsspaton: t u c S c f (9) 1 t 2 2 t x x k k Besdes, all the model terms/symbols/coeffcents/functons have ther usual meanngs and values Hybrd RANS-LES Turbulence Modellng The Spalart Allmaras turbulence model s a one-equaton model for the eddy vscosty. The use of ths model s otherwse known as Hybrd RANS-LES modellng or Detached Eddy Smulaton (DES) modellng. The dfferental equaton s derved by usng emprcsm and arguments of dmensonal analyss, Gallean nvarance and selected dependence on the molecular vscosty. Grd resoluton does not need to be fner for ths model, however, one can essentally apprehend the velocty feld gradent wth the assocated algebrac models. The transport equaton for the workng varable (otherwse termed as Spalart Allmaras varable).e. vscosty-lke varable (ṽ) s expressed as follows: 2 ~ ~ 1 ~ ~ ~ ~ ~ ~ ~ ~ u cb 1S cb2 c 1 f (10) w w t x x x x x d The eddy vscosty can be expressed as follows: ~ t fv1 t (11) Furthermore, all the model terms/symbols/coeffcents/functons have ther usual meanngs and values. V. NUMERICAL PROCEDURES 5.1. Numercal scheme and soluton algorthm The above-mentoned governng transport equatons are converted nto a much generalsed form as mentoned:. u. u S (12) t The converted governng transport equatons are dscretzed by expendng a pressure based coupled framework relatng to fnte volume method (FVM) usng the SIMPLER algorthm, where ϕ represents any conserved varable and S s a source term. The establshed pressure based, All Rghts Reserved 181

5 coupled solver s used to predct flow behavours of the related flow varables n connecton wth supersonc turbulent flow over a backward facng sharp edge step Choce of grd sze, tme step and convergence crtera Fgure 3 demonstrates that the grd of the computatonal doman s consdered to be non-unform and also grd s refned near the vcnty where the hgh gradent s expected.. In the present work, the smulaton of both the turbulence models wth dfferent wall dstance from grd s carred out on the computatonal doman. A comprehensve grd-ndependence test s performed to establsh a sutable spatal dscretzaton, and the levels of teraton convergence crtera to be used. As an outcome of ths test, we have used non-unform grds for the fnal smulaton. Correspondng tme step taken n the smulaton s seconds. Though, t s checked wth smaller grds of n numbers, t s observed that a fner grd system does not alter the results sgnfcantly. Convergence n nner teratons s declared only when the condton s satsfed smultaneously for all varables, where φ stands for the feld varable at a grd pont at the current teraton level, φold represents the correspondng value at the prevous teraton level, and φmax s the maxmum value of the varable at the current teraton level n the entre doman. Fgure 3. Mesh for backward facng sharp edge step VI. RESULTS AND DISCUSSIONS Wth the prevously mentoned model condtons, the numercal smulatons are accomplshed for examnng the flud flow behavours of the accompanyng flow varables concernng about the supersonc turbulent flow over a backward facng sharp edge step Grd ndependences test The grd of the computatonal doman s consdered to be non-unform and also grd s refned near the vcnty where the hgh gradent s expected to capture the flow physcs. Three nconsstent grd types namely 1, 2 and 3 of ( ), ( ) and ( ), respectvely, are used for the smulaton n order to perform the grd dependence test for gettng optmal grd. The optmal grd s found to be grd 2 of ( ), for the present computatonal doman, whch s used for all future nvestgatons. The detaled grd specfcatons and the grd dependence test results are demonstrated n table 1 and fgure 4, respectvely, for the present case of sharp edge backward All Rghts Reserved 182

6 Table 1. Detaled grd specfcatons for sharp edge backward step Name Number of Ponts Number of Cells Number of Faces Grd Grd Grd Fgure 4. Grd ndependence test 6.2. Valdaton of numercal predctons aganst expermental results The smulaton results have been valdated aganst the correspondng avalable expermental data of Smth [1], so that the smulaton accuracy can be predcted n advance. The models represented for valdaton are both RANS and hybrd RANS-LES models and also predct the relatve accuracy for the present smulaton. The sample locaton wthn the smulaton flow feld, for valdatng the work wth expermental data s represented by the whte lne as shown n fgure 5. The valdaton for pressure recovery plotted aganst the assocated expermental data s llustrated n fgure 6. It shows that the accuracy of the RANS model s lmted to near wall regon and the hybrd RANS-LES mantan the consstency n accuracy beyond the wall vcnty. Therefore, n overall, the hybrd RANS-LES model gves more accurate results than the RANS model. Hence, the hybrd RANS-LES s consdered for all further examnatons. Fgure 5. Pressure recovery dstrbuton for steady state All Rghts Reserved 183

7 6.3. Comparson of numercal predctons wth expermental results The smulaton accuracy s hghly dependent on grds and non-dmensonal sublayer-scaled dstance y +.e. uty/ν. Fgure 7 demonstrates the effects of varous y + on smulaton accuracy of the RANS- LES turbulent model plotted aganst the related expermental data. Even though, for capturng velocty gradents fne grd resoluton s not necessary for hybrd RANS-LES model, however, the non-unform structured grd wth refnement n the vcnty of expected hgh gradent s very much sutable and approprate for better solver convergence and resoluton near the wall. That s why, n the present nvestgaton the refnement near the wall and the separaton lne s mantaned wth y + of 1 and 11 and the correspondng results are as shown n fgure 7. The computatonal tme for y + = 1 s much hgher than y + = 11. However, y + = 1 wth the hybrd RANS-LES has less computatonal tme as compared to Drect Numercal Soluton (DNS) and also both converge wth the related expermental data through the same accuracy. Fgure 6. Pressure recovery dstrbutons valdated wth expermental data Fgure 7. Effect of y + on smulaton All Rghts Reserved 184

8 VII. CONCLUSIONS In the current research work, a two dmensonal numercal model s establshed for flow over a backward facng sharp edge step ntroducng both RANS and hybrd RANS-LES turbulent models. The models also takes nto account the dynamc elements lke producton, dffuson and destructon terms along wth the natural facets conformng wth the present research topc. The numercal smulatons are conducted wth the ndcated models by consderng the nflow free stream Mach number of 2.5 assocated wth free stream pressure and velocty of N/m 2 and m/s 2, respectvely. Above and beyond, the grd dependence test s carred out for gettng the optmal grd, whch s observed to be of ( ), n the present case. The smulaton results are also compared wth the correspondng expermental results exstng n the lterature. Even though both the model results are congenal, stll, the hybrd RANS-LES model yelds comparatvely better and consstent results over the RANS model wthn the whole flow doman. Furthermore, for better resoluton, smulaton accuracy and solver convergence proxmate to the wall vcnty, the non-dmensonal sublayer-scaled dstance y + of 1 and 11 are taken n the current studes wth hybrd RANS-LES. Also, the sudden expanson flow over the sharp edge step s the man cause of the uneven flow behavors as observed from the pressure recovery dstrbutons. ACKNOWLEDGMENTS The author would lke to thank the edtor and the revewers for ther noble thoughts, valuable tme and esteemed contrbutons for gvng nsghtful revews to the research artcle. REFERENCES I. Smth, Howard E. The flow feld and heat transfer downstream of a rearward facng step n supersonc flow. No. ARL Aerospace Research Labs, Wrght Patterson AFB, Oho, (1967). II. Launder, B. E., and B. I. Sharma. "Applcaton of the energy-dsspaton model of turbulence to the calculaton of flow near a spnnng dsc." Letters n heat and mass transfer Vol. 1, Issue 2 (1974): III. Armaly B. F., Durst F., Perera J. C. F., and Schoenung B., Expermental and theoretcal nvestgaton of backward facng step flow, Journal of Flud Mechancs, Vol. 127, pp , (1983). IV. Spalart, Phllpe R., and Steven R. Allmaras. "A one-equaton turbulence model for aerodynamc flows." (1992). V. Anderson, John Davd, and J. F. Wendt. Computatonal flud dynamcs. Vol New York: McGraw-Hll, (1995). VI. Neumann, Jens, and Hans Wengle. "DNS and LES of passvely controlled turbulent backward-facng step flow." Flow, turbulence and Combuston (2003): VII. Hamed, A., K. Das, and D. Basu. "Numercal smulatons of fludc control for transonc cavty flows." AIAA Paper 429, (2004). VIII. Chen, Zh, et al. "An expermental study on fne structures of supersonc lamnar/turbulent flow over a backwardfacng step based on NPLS." Chnese Scence Bulletn, Vol. 57, Issue 6 (2012): IX. Lu, Haxu, et al. "Effects of Inflow Mach Number and Step Heght on Supersonc Flows over a Backward- Facng Step." Advances n Mechancal Engneerng (2013). X. V. I. Terekhov, Ya. I. Smul sk, and K. A. Sharov, Expermental study of the separated flow structure behnd a backward-facng step and a passve dsturbance, Journal of Appled Mechancs and Techncal Physcs, Volume 57, Issue 1, (2016) pp 180 All Rghts Reserved 185