Simulation of Turbulent Flow Using FEM

Transcription

1 Internatonal Journal of Engneerng and Technology Volume 2 No. 8, August, 2012 Smulaton of Turbulent Flow Usng FEM Sabah Tamm College of Computng, AlGhurar Unversty, Duba, Unted Arab Emrates. ABSTRACT An analyss of developng and fully developed turbulent flow was carred out when dfferent technques have been used n a zone close to the sold wall. In ths paper, wall element technque based on the fnte element method (FEM) has been adopted and compared wth other technques for determnaton of confned turbulent flow wth a one equaton model used to depct the turbulent vscosty s appled to a smooth straght channel. Keywords: Developng and fully developed turbulent flow, Pressure flow, Coupled and teratve methods. 1. INTRODUCTION Flud dynamcs has many mportant applcatons. The varetes of these applcatons represented n wde range of appled computer scence as well as engneerng problems. Due to the growth of technology, these applcatons take advantage of the ncreasng speed of computers and hardware capabltes. The flud moton can be represented by the well known Naver-Stokes (N-S) equatons. Due to the complety of these equatons an analytcal soluton s ntractable and the last three decades, wth the development and avalablty of more powerful dgtal computers much attenton has been focused on the numercal smulaton for solvng the resultng set of non lnear partal equatons whch domnate the flow behavor processes, the so-called computatonal flud dynamcs (CFD). Ths has been developed and used wth confdence to solve a large range of flow problems and at the same tme can offer a cost-effectve to many flud problems. Indeed, under condtons where epermentaton s etremely dffcult, CFD may be the only methodology avalable. Numerous theoretcal and epermental works are avalable on lamnar flow [1-2], but ths s not the case of turbulent flow. Snce t has not been possble to obtan eact analytcal solutons to such flows, an accurate numercal approach would be very benefcal to researchers. The fnte element method (FEM) s one of these methods that have recently emerged as a powerful tool for solvng the N-S equatons. Wthn the computatonal doman (.e. man doman), the fnte element method s used to dscretse the equatons governng the flud moton. It s well known that when a flud enters a prsmodal duct the values of the pertnent varables change from some ntal profle to a fully developed form, whch s thereafter nvarant n the downstream drecton. The analyss of ths regon, whch s known as developng regon, has been the subect of etensve studes. An effectve technque s requred to model the varaton of the pertnent varables near a sold boundary, where the varaton n velocty and knetc energy, n partcular, s etremely large near such surfaces snce the transfer of shear form the boundary nto the man doman and the nature of the flow changes rapdly. A sgnfcant grd refnement would be requred, f a conversatonal fnte element s used to model the near wall zone (N.W.Z.). Therefore, several soluton technques have been suggested n order to avod such ecessve refnement [3-5]. A more common approach s to termnate the actual doman subect to dscretsaton (man doman) at some small dstance away from the wall, where the gradents of the ndependent varables are relatvely small, and then a technque s a requred to model the flow behavor n the near wall element. In ths paper, dfferent technques were used. One of them was the use of an element technque, whch s based on the use of the one dmensonal fnte element technque (.e. one dmensonal element n one drecton normal to the sold wall). The valdty of the wall element technque has been tested and proved n the prevous work for fully developed turbulent flow [6-7]. Presently, the valdty of the technque has been tested for developng flow, along wth other technques to smulate turbulent flow n a smooth straght channel. 2. MATHEMATICAL DESCRIPTION The current nvestgaton relates to steady - state ncompressble two dmensonal turbulent flow of a Newtonan vscous flud wth no body forces actng. For such a stuaton, the Naver- Stokes (N-S) equatons assocated wth ths type are, u u p = - + u e u... (1) ISSN: IJET Publcatons UK. All rghts reserved. 1358

2 Internatonal Journal of Engneerng and Technology (IJET) Volume 2 No. 8, August, 2012 Where,= 1,2. u, p are the tme - averaged veloctes and pressure respectvely, s the flud densty, e s the effectve vscosty whch s gven by e = + t, and t are the molecular vscosty and turbulent vscosty, respectvely. The flow feld must satsfy the contnuty equaton, whch may be wrtten as: u = 0.. (2) Equaton (1) and (2) cannot be solved unless a turbulence closure model can be provded to evaluate the turbulent contrbuton to e. In the present work, a one equaton model has been adopted so that, Ck 1 1/ 2 t... (3) 1 s the length scale of turbulence whch s gven by 1 = m, 1 m s the mng length based on the prandtl hypothess whch has been specfed algebracally for the present purposes as 0.4 tmes the normal dstance from the nearest wall surface, C s a constant and k s the tme-averaged turbulence knetc energy. The t gven by equaton (3) requres that k to be known. Ths can be evaluated va a further transport equaton gven by: k k u u u t u t E.. (4) k 2 Where E= C 3/ D k / 1, t / s the turbulent dffuson k coeffcent, k s the turbulent prandtl or Schmdt number and C D s a constant. The turbulence model based on equatons (1),(2) and (4) are called the one-equaton (k-l) model. The above governng equatons have been solved usng a fnte element method [8]. Wthn the near wall zone ether unversal laws concept [9], or one dmensonal parabolc elements n a drecton normal to sold wall s adopted. Wthn the man doman, conventonal two dmensonal soparametrc elements doman (.e. 2-D up to the wall), are used to dscretse the flow doman.. In order to avod such ecessve refnement, sememprcal equatons, known as Wall Functons or the so-called Unversal Laws, are used to brdge from a sold boundary to the man doman.. In the present work, a fnte elements technque has been adopted, usng one-dmensonal (3-noded elements) normal to the wall as shown n Fgure 2. In () the momentum equatons n drecton normal to the wall surface, together wth pressure equaton and the knetc energy equaton are solved n the near wall zone. 4. BOUNDARY CONDITIONS The boundary condtons appled are as follows: 1. Veloctes and pressure. Forced boundary condtons such as, on the boundary 1 ( velocty ). Tracton boundary condtons, where the tracton s are ether defned or updated on boundary, e u1 p 1 - parallel to walls 1 1 e u 2 u normal to walls 2 2. Turbulent knetc energy (k) as per () above or updated Neumann condtons. In ths work, posulle flow s consdered only and the boundary condtons were mposed as shown n Fgure 1. Compatble fully developed velocty and knetc energy profles whch look lke parabolc curve were mposed at the upstream secton when fully developed turbulent flow was consdered at the frst stage and the tracton s were updated at downstream. These profles were obtaned by usng the outlet values form each teraton as new appromatons to the values at the nlet untl a convergent condton s satsfed. 3. NEAR WALL ZONE TREATMENT Wthn the near wall zone (Fgure 1) dfferent technques were used, these are as follows,. Conventonal fnte elements (.e. 2-D elements up to the wall) are used to dscretse the N.W.Z. and the varable values, followng analyss, are used as reference data. However an ecessve mesh refnement was needed whch s epensve n computer tme and memory. Fgure 1: Boundary condtons when the mesh s termnated at small dstance away from the wall ISSN: IJET Publcatons UK. All rghts reserved. 1359

3 Internatonal Journal of Engneerng and Technology (IJET) Volume 2 No. 8, August, 2012 of 2-D elements up to the wall s not economcally vable snce t needs an ecessve refnement whch s very costly n computer tme and memory sze. The wall element technque has been appled successfully to fully developed flow. Snce each 1-D strng of elements s analysed ndvdually, ths saved computer memory and tme requred. For eample, converged soluton was obtaned n 71 sec when 1-D elements n one drecton employed; and 193 sec when unversal laws employed. However, a consderable savng n both can result f a comparson s made wth analyss when 2-D elements are used up to the wall The second stage was concerned wth developng turbulent flow n a channel. The velocty mposed upstream was constant at 2.0 m/sec and the knetc energy mposed as 0.02 m 2 /sec 2 and tractons updated downstream. The results are assumed to be converged when the relatve change n any varables s less than 1%. A very fne mesh dstrbuton was used such that wth further refnement no ncrease n a accuracy was apparent. Fgure 2: One-dmensonal elements n one-drecton normal to the wall used n the N.W.Z Converged velocty profles for developng turbulent flow are llustrated n Fgure 7, whch shows that the results obtaned from the adopted wall element technque corresponds to that when the whole doman s mapped but dffer from those when the unversal laws was used. 5. RESULTS AND DISCUSSION The valdty of the adopted wall element technque (1-D elements n one drecton) has been tested for two stages developng and fully developed flow, and compared wth other accepted technques and wth epermental results n a parallel-sded duct of wdth D, whch s taken as 1.0 n the present work, and length L. Dfferent Reynolds number based upon the wdth of the channel of and were consdered when pressure flow was consdered only. In the frst stage, Full developed turbulent flow was consdered, n ths compatble fully developed velocty and knetc energy profles were mposed as ntal upstream values. These values obtaned by usng the outlet values as nlet for the net teraton untl a converged condton s satsfed. Fgures 3 clearly shows that the velocty values obtaned by unversal profles have some dscrepancy from those obtaned from the advocated wall element technque (.e. 1-D n 1 drecton normal to the wall), and also shows an ecellent agreement wth the correct soluton whch resulted from the complete mappng (.e. 2-D up to the wall). In fact, these are, superor to those obtaned usng unversal laws. Fgure 4 shows ecellent agreement between the adopted technque and epermental results [10]. Fgures 5 and 6 refer to the knetc energy and the turbulent vscosty. These clearly demonstrate that the results obtaned from the adopted wall element technque (1-D elements n one drecton) and 2-D elements up to the wall are dentcal. The concluson of ths stage s that, the wall functons (unversal laws) have been shown to have lmted applcaton. Also, the use The velocty, knetc energy and turbulent vscosty profles presented n Fgures 8,9 and 10, ndcates that for ths case the correspondence between the results obtaned when full mappng s used and 1-D element n one drecton are not as good as that eperenced prevously, compared to those obtaned n fgures 5 and 6. The concluson of ths stage s that, the use of the unversal laws technque s not acceptable any longer for developng flow and the use of 2-D elements up to the wall s not economcally vable. Also, the accuracy of the wall element technque when used n one drecton s clearly not vald for developng flow. The fnal stage was concerned wth etenson of NWZ further nto man doman n order to prove the valdty of the technque when developng flow s consdered. Agan, two dfferent methods of solutons (coupled and uncoupled) were used to obtan the velocty and knetc energy as presented n Fgures 11, 12 and 13 when 1-D elements normal to the wall are employed n the NWZ. Clearly, Fgure 12, the velocty plots became smoother when all element types are emboded wthn one overall matr, especally when the mesh s termnated at 0.47D as compared to the curves n Fgure 11 when an uncoupled method based on an teratve technque was employed. Fgure 13 confrmed that the use of coupled method gave smoother results when he NWZ was etended up to 0.47D and correspondng to Y + = 31 at ths locaton. The concluson of the fnal stage s that, the use of coupled method was better than the teratve method when the NWZ was etended. Ths apples to both developng and fully-developed flow. However, for low Y + values the uncoupled method was stll applcable. ISSN: IJET Publcatons UK. All rghts reserved. 1360

4 Internatonal Journal of Engneerng and Technology (IJET) Volume 2 No. 8, August, 2012 Fgure 3: Turbulent velocty profles for fully-developed flow, at 8D downstream, L=8D, Re= Fgure 6: Vscosty dstrbuton profles for fully-developed turbulent flow, at 8D downstream, L=8D, Re= Fgure 4: Turbulent velocty profles for fully-developed flow, at 8D downstream, L=8D, Re= Fgure 7: Developng velocty profles for turbulent flow, at 10D downstream, L=10D, Re= Fgure 5: Knetc energy profles for fully-developed turbulent flow, at 8D downstream, L=8D, Re= Fgure 8: Developng velocty profles for turbulent flow, at 10D downstream, L=10D, Re= ISSN: IJET Publcatons UK. All rghts reserved. 1361

5 Internatonal Journal of Engneerng and Technology (IJET) Volume 2 No. 8, August, 2012 Fgure 9: Developng knetc energy profles for turbulent flow, at 10D downstream, L=10D, Re= Fgure 12: Developng velocty profles for turbulent flow, at 10D downstream, L=10D, Re= Equatons solved n one matr. Fgure 10: Developng Vscosty dstrbuton profles for turbulent flow at 10D downstream, L=10D, Re= Fgure 13: Developng knetc energy profles for turbulent flow, at 10D downstream, L=10D, usng 1-D n one Dr, Re=12.000, at nterface 0.47D 6. CONCLUSIONS Fgure 11: Developng velocty profles for turbulent flow, at 10D downstream, L=10D, Re= Equatons solved usng teratve technque. 1. The use of the unversal laws technque s not acceptable any longer for ether developng or fully developed flow. 2. The general use of 2-D elements up to the wall s not economcally vable. Therefore to avod such an ecessve refnement, these methods have been replaced by ntroducng a wall element technque, based on the use of the fnte element methods. ISSN: IJET Publcatons UK. All rghts reserved. 1362

6 Internatonal Journal of Engneerng and Technology (IJET) Volume 2 No. 8, August, The adopted wall element technque has been appled successfully and proved to be superor to other technques and, can be used wth confdence for fully developed turbulent flow. Therefore, ths technque s vald for fully developed flow but not for developng flow snce the assumpton of undrectonal flow s unacceptable. Therefore, ths technque can be used wth confdence for turbulent fully-developed flow only, but not for developng flow. 4. When the NWZ was etended. The use of coupled method shows better and smoother results than those obtaned when the teratve method was used. Ths apples to both developng and fully-developed flow. REFERENCES [1] D.M. Hawken, H.R. Tamaddon-Jahrom, P.Townsend and M. F. Webster, A Taylor-Galerkn based algorthm for vscous ncompressble flow, Int. Journal Num. Meth. Fluds, (1990) [2] Mehrotra, A. K. and Patence. G. S., Unfed Entry Length for Newtonan and power law fluds n Lamnar ppe flow, J. Chem. Eng., Vol. 68, pp , (1990) [3] Launder, B. E. and Shma, N., Second moment closure for near wall sublayer: Development and Applcaton, AIAA Journal, Vol. 27, pp , (1989) [4] Haroutunan, and Engelman, S., On modelng wallbound turbulent flows usng specalzed near-wall fnte elements and the standard k- turbulent model. Advances n Num. smulaton of Turbulent flows, ASME, Vol. 117, pp , (1991) [5] Graft T, Gerasmov A, Iacovdes H, Launder B., Progress n the generalzaton of wall-functon treatments, Int. Journal for heat and flud flow. [6] P , (2002) [7] Sabah Tamm, Valdaton of wall element technque of turbulent flow, GSTF Internatonal Journal on Computng (JoC), Volume 2, No. 1, pp , (2012). [8] Sabah Tamm, An Effectve Technque Usng Fnte Element Methods, Internatonal Journal of Electrcal & Computer Scence IJECS-IJENS, Vol.12, No. 2, pp. 1-5, (2012). [9] Taylor, C. and Hughes, T. G., Fnte element programmng of the Naver-Stokes equaton, Pnerdge press, (1981) [10] Daves, T. J., Turbulent phenomena, Academc Press,(1972) [11] Nayak, U.S.L. and Stevens, S.J, An epermental study of the flow n the annular gap between a long vehcle and a low close-fttng tunnel, Report: Dept. of Technology, Loughborough Unversty of Technology, (1973) ISSN: IJET Publcatons UK. All rghts reserved. 1363