Numerical Studies on Supersonic Turbulent Flow over a Backward Facing Sharp Edge Step Using Hybrid RANS-LES

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1 Numercal Studes on Supersonc Turbulent Flow over a Bacward Facng Sharp Edge Step Usng Hybrd RANS-LES Dr. Nrmal Kumar Kund Assocate Professor, Department of Producton Engneerng, Veer Surendra Sa Unversty of Technology, Burla , Inda Abstract In the present study, a two dmensonal numercal model s developed to nvestgate supersonc turbulent flud flow over a bacward facng sharp edge step by usng both RANS (relatng to the standard -ε model) and hybrd RANS-LES (pertanng to the Spalart-Allmaras model nvolvng vscostyle varable) models. The model also nvolves addtonal mportant factors namely producton, dffuson and destructon terms besdes the very common aspects related to the present research problem. The numercal smulatons are performed usng the stated turbulence models wth the nflow free stream Mach number of.5 along wth the free stream pressure and velocty of N/m and m/s, respectvely. The smulaton predctons are compared wth the correspondng expermental results avalable n the lterature. Even though the smulaton predctons from both RANS and hybrd RANS-LES models yeld satsfactory results, however, the hybrd RANS- LES model gves farly better and consstent results over the RANS model (whch provdes very good results wthn the vcnty of the wall regon only) throughout the entre flow regon and hence, only the hybrd RANS-LES s consdered for further nvestgatons. It s also notced that the sudden vscous layer separaton s the prme cause for the generaton of reattachment shoc. In addton, the uneven pressure recovery s owng to the sudden expanson flow over the sharp edge step. Further, the sudden expanson flow ncreases the ntensty of the shoc whch leads to uneven flow behavours. Defntely, the present test case results are very hghly benefcal to understand the flow characterstcs of supersonc turbulent flud flow over the bacward facng sharp edge step flows. Index Terms Supersonc, Turbulent Flow, Bacward Facng, Sharp Edge Step, RANS, Hybrd RANS-LES 1. INTRODUCTION The flow over bacward-facng step s one of the mportant framewors and has ganed specfc attenton on account of not ust smplcty but for volumnous technologcal applcatons. 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 bacward 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 mae the flow regme more complex and hence momentous nvestgatons are very much requred for mprovng the lvely desgn of arcrafts.. LITERATURE REVIEW Smth [1] carred out expermental nvestgatons on the flow feld and heat transfer downstream of a rearward facng step n supersonc flow. Launder and Sharma [] 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 bacward 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 bacward-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 bacward-facng step by usng Nano-based Planar Laser Scatterng (NPLS). Lu et al. [9] nvestgated numercally on the nfluences of nflow Mach number and step heght on supersonc flows over a bacwardfacng step. Terehov et al. [10] performed the expermental studes on the separated flow structure behnd a bacwardfacng step n addton to the passve dsturbance. From the stated nvestgatons, to the best of author nowledge, t s found that there s not a sngle comprehensve numercal study on flow over a bacward facng sharp edge step usng hybrd RANS-LES method. Wth ths perspectve, the present research demonstrates the numercal studes on flow behavours over a bacward facng sharp edge step usng hybrd RANS-LES technque. In addton, the numercal model also nvolves addtonal mportant features namely producton, dffuson and destructon terms besdes the common ssues relatng to the present physcal problem. Furthermore, the stated model also ncludes both compressblty and eddy vscous effects. The model s very well demonstrated for the metculous numercal ISSN: EverScence Publcatons 14

2 studes on flud flow characterstcs pertanng to flow over a bacward facng sharp edge step by ntroducng the nflow free stream velocty along wth the correspondng Mach number as the ey model parameters. Ultmately, the present case of bacward facng sharp edge step for both RANS (assocated wth the standard -ε model) and hybrd RANS- LES (concernng Spalart-Allmaras model nvolvng vscostyle varable) predctons of fully supersonc turbulent flow are compared wth expermental data of Smth [1]. Eventually, the model predctons wth regard to the specfed ey model parameters are also along the expected lnes and are n very good agreement wth the correspondng expermental results. 3. DESCRIPTION OF PHYSICAL PROBLEM Bacward facng sharp edge step havng wde range of applcatons n appled aerodynamcs s nvestgated n the present research. The geometrc confguraton along wth ntal and boundary condtons are referred from the expermental research report of Smth [1] Geometrc Model Fgure 1 represents the setup confguraton for testng the bacward facng sharp edge step flow over sharp edge geometry separatng at a step heght H = m, upstream dstance from nlet to step L u = m and downstream dstance from sharp edge step to outlet L d = 0.03 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 represented by S and R respectvely and are expected to be observed after performng numercal smulaton. Fg 1. Flow specfcaton of bacward facng sharp edge step 3.. Intal and Boundary Condtons The nflow free stream velocty U n = m/s, for whch the nown statc free stream pressure p n = N/m corresponds to the Mach number Ma =.5. At the left sde ahead of the step, the ntal temperature s mantaned at 169. K. The ntal condtons whch are set on the upstream are very much useful throughout the smulaton along the spanwse drecton, for gettng the flow characterstcs beyond the step. For the turbulence, both RANS standard -ε two-equaton model and Spalart-Allmaras one-equaton hybrd RANS-LES (otherwse termed as Detached Eddy Smulaton, DES) model are taen nto account. The boundary condtons for the geometry represented by fgure are as follows: Pressure p = Pa, everywhere else for pressure n case of both RANS and hybrd RANS-LES models. Temperature T n = 169. K, everywhere else for temperature for both RANS and hybrd RANS-LES models. Velocty U n = 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. Fg. Bacward facng sharp edge step boundary representaton 4. MATHEMATICAL FORMULATION The most generalzed governng transport equatons of mass, momentum and energy expressed n the conservatve form of Naver-Stoes equaton for compressble flow accompanyng the nfluences of turbulence are as mentoned underneath. Contnuty: ( u t x Momentum: u t Energy: ) 0 ( uu x ) p x x (S ) t (1) () ISSN: EverScence Publcatons 143

3 E t x Where, u u p p T T x u E p T x t S t u Sh u p T Total energy, p v E e h The Reynolds stress term s modeled n terms of the eddy vscosty and s expressed as: t ( S S / 3) / 3 (6) t nn The eddy vscosty s defned as a functon of the turbulent netc energy, and the turbulent dsspaton rate ε, and s expressed as: c f t / (7) In addton, all the model terms / symbols / coeffcents / functons have ther usual meanngs and values RANS Turbulence Modellng The standard -ε model s the most wdely nown and used extensvely for two-equaton eddy vscosty model. Transport equatons are nterpreted by two scalar propertes of turbulence.e., the transport equaton -equaton s a model for the turbulent netc energy and the ε-equaton s a model for the dsspaton rate of turbulent netc energy. The turbulent transport equatons for the -ε model are defned as follows: Turbulent netc energy: t t u t x x S Energy dsspaton: (3) (4) (5) (8) c 1 t x ts t u x c f Besdes, all the model terms/symbols/coeffcents/functons have ther usual meanngs and values. 4.. 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 nown 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 worng varable (otherwse termed as Spalart Allmaras varable).e. vscosty-le varable (ṽ) s expressed as follows: ~ ~ ~ u t x (10) ~ ~ ~ 1 ~ ~ ~ ~ cb 1S cb cw 1 fw x x x x d The eddy vscosty can be expressed as follows: ~ t fv 1 t (9) (11) Furthermore, all the model terms / symbols / coeffcents / functons have ther usual meanngs and values. 5. NUMERICAL PROCEDURES 5.1. Numercal scheme and soluton algorthm The aforesad governng transport equatons are converted nto much generalzed form as follows.. u. u S (1) t The converted governng transport equatons are dscretzed by expendng a pressure based coupled framewor 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, fully coupled solver s used to predct flow behavours of the related flow varables ISSN: EverScence Publcatons 144

4 n connecton wth supersonc turbulent flow over a bacward facng sharp edge step. 5.. Choce of grd sze, tme step and convergence crtera Fgure 3 shows 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 wor, 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 taen n the smulaton s seconds. Though, t s checed wth smaller grds of n numbers, t s observed that a fner grd system does not alter the results sgnfcantly Comparson of numercal predctons wth expermental results The numercal smulaton predctons are conscentously compared wth the avalable expermental data of Smth [1] reported n the lterature, n order to ascertan the smulaton accuracy beforehand. The turbulence models taen nto account (for the present case of fully supersonc turbulent flud flow over the bacward facng sharp edge step) for comparsons of the numercally predcted results (wth the correspondng expermental data) are both RANS (relatng to the standard -ε model) and the hybrd RANS-LES (assocated wth the Spalart-Allmaras model nvolvng the vscosty-le varable (ṽ)). The model (between RANS and RANS-LES) renderng the relatvely better smulaton accuracy wll be consdered for further nvestgatons. Fg 4. Pressure recovery comparson wth expermental data Fg 3. Mesh for bacward facng sharp edge step The convergence n nner teratons s declared only when the condton φ φ old φ max 10 4 s satsfed smultaneously for all varables, where φ stands for the feld varable at a grd pont at the current teraton level, φ old represents the correspondng value at the prevous teraton level, and φ max s the maxmum value of the varable at the current teraton level n the entre doman. 6. RESULTS AND DISCUSSIONS Wth the already descrbed model condtons, the numercal smulatons are performed for nvestgatng the flud flow behavours of the assocated flow varables pertanng to supersonc turbulent flow over a bacward facng sharp step. The comparsons of the numercal smulaton research wors for pressure recovery wth the expermental data avalable n the lterature reveals that the accuracy of the RANS model s lmted to near wall regon, whereas, the hybrd RANS-LES mantan the consstency n accuracy beyond the wall vcnty as demonstrated n fgure 4. Therefore, the hybrd RANS-LES model gves rse to relatvely superor and precse predctons than RANS model. Hence, only the hybrd RANS-LES s taen nto consderatons for further studes. 6.. Pressure dstrbutons Fgure 5 llustrates the coloured pressure contour along wth the vertcal scale bar, representng the decrease n pressure near the vcnty of the expanson fan regon, whereas, the reattachment shoc regon has experenced more pressure gradent. In addton, the recrculaton vcnty whch s also nown as dead ar regon has experenced the least pressure ISSN: EverScence Publcatons 145

5 because of non-vscous rotaton. Furthermore, the supersonc turbulent flow over the bacward facng sharp edge step has also experenced the notceable pressure fluctuatons between the expanson fan and the reattachment shoc wave regons. Addtonally, t s qute evdent that owng to the shoc generaton pressure recovery behnd the sharp edge step s also not mmaculate enough for smooth and flawless flud flow. In addton, the physcs behnd the pressure gradent caused by two parallel shocs may easly be understood from the coloured pressure feld together wth the vertcal scale bar, as demonstrated n fgure 6. approaches to the bottom wall whch wll follow along the ntal drecton. However, a part of the flow reverses to the dead ar regon whch causes t as a recrculaton regon. Furthermore, ths crculaton causes the ncreasng n the shear layer length whch leads to flow feld losses. In order to obtan the qute smooth and flawless flow the recrculaton vcnty has to be mnmzed or elmnated. In addton, the coloured velocty vector along wth the horzontal scale bar, wthn the flud flow regon s very well demonstrated n fgure 8, whch benefts for very clear understandng of the reattachment pont flow physcs near the bottom wall. Fg 5. Pressure contour Fg 7. Velocty profle along spanwse drecton 6.3. Velocty dstrbutons Fg 6. Pressure feld Fgure 7 depcts the velocty profle along the spanwse drecton relatng to the supersonc turbulent flud flow over the bacward facng sharp edge step. It s qute apparent that the velocty accelerates due to sudden expanson over the sharp edge step leadng to vscous layer separaton and formaton of shear layer whch approaches to the bottom wall resultng n reattachment shoc and redevelopment of boundary layer. In addton, the flow around the shear layer Fg 8. Velocty vector wthn flud flow doman ISSN: EverScence Publcatons 146

6 7. CONCLUSIONS A two dmensonal numercal model s establshed to study the fully supersonc turbulent flud flow characterstcs over a bacward facng sharp edge step. The model also nvolves addtonal ey factors le producton, dffuson and destructon terms besdes the very usual aspects pertanng to the present physcal research problem. The model s very well demonstrated for the panstang numercal nvestgatons on flud flow behavours wth both the turbulent compressble flud flow models namely RANS (assocated wth the standard -ε model) and hybrd RANS-LES (relatng to Spalart-Allmaras model nvolvng vscosty-le varable (ṽ)) by ntroducng the nflow free stream velocty along wth the correspondng Mach number as the ey model parameters. The smulaton predctons from both the stated turbulent models are compared wth expermental data avalable n lterature. Eventually, the model predctons wth regard to the specfed ey model parameters are also along the expected lnes and are n very good agreement wth the correspondng expermental results. However, the hybrd RANS-LES model s found to gve reasonably better accuracy than the RANS model for the same computatonal cost and hence, only the former/frst one s consdered for all other studes. It s observed that the sudden vscous layer separaton s the man reason for the generaton of reattachment shoc. Furthermore, the uneven pressure recovery s caused by sudden expanson flow over the sharp edge step. Besdes, the sudden expanson flow ncreases the ntensty of the shoc whch results n uneven flow characterstcs. The development of recrculaton regon leads to shear layer formaton and the reattachment length s also found n the sharp edge step. In addton, the numercal modellng for supersonc turbulent flud flow over a bacward facng round step s underway and s planned for the future to reduce recrculaton regon whch can result n smaller shear layer formaton and the shorter reattachment length, for the very smlar test case condtons. REFERENCES [1] 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). [] 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 (1974): [3] Armaly B. F., Durst F., Perera J. C. F., and Schoenung B., Expermental and theoretcal nvestgaton of bacward facng step flow, Journal of Flud Mechancs, Vol. 17, pp , (1983). [4] Spalart, Phllpe R., and Steven R. Allmaras. "A one-equaton turbulence model for aerodynamc flows." (199). [5] Anderson, John Davd, and J. F. Wendt. Computatonal flud dynamcs. Vol. 06. New Yor: McGraw-Hll, (1995). [6] Neumann, Jens, and Hans Wengle. "DNS and LES of passvely controlled turbulent bacward-facng step flow." Flow, turbulence and Combuston (003): [7] Hamed, A., K. Das, and D. Basu. "Numercal smulatons of fludc control for transonc cavty flows." AIAA Paper 49, (004). [8] Chen, Zh, et al. "An expermental study on fne structures of supersonc lamnar/turbulent flow over a bacward-facng step based on NPLS." Chnese Scence Bulletn, Vol. 57, Issue 6 (01): [9] Lu, Haxu, et al. "Effects of Inflow Mach Number and Step Heght on Supersonc Flows over a Bacward-Facng Step." Advances n Mechancal Engneerng (013). [10] V. I. Terehov, Ya. I. Smul s, and K. A. Sharov, Expermental study of the separated flow structure behnd a bacward-facng step and a passve dsturbance, Journal of Appled Mechancs and Techncal Physcs, Volume 57, Issue 1, (016) pp ISSN: EverScence Publcatons 147