Fourier time spectral method for subsonic and transonic flows

Transcription

1 Act Mech. Sin. DOI.7/s x RESEARCH PAPER Fourier time spectrl method for susonic nd trnsonic flows Lei Zhn, Feng Liu Dimitri Ppmoschou Received: Septemer 5 / Revised: 3 Octoer 5 / Accepted: Novemer 5 The Chinese Society of Theoreticl nd Applied Mechnics; Institute of Mechnics, Chinese Acdemy of Sciences nd Springer-Verlg Berlin Heidelerg 6 Astrct The time ccurcy of the exponentilly ccurte Fourier time spectrl method (TSM) is exmined nd compred with conventionl nd-order ckwrd difference formul (BDF) method for periodic unstedy flows. In prticulr, detiled error nlysis sed on numericl computtions is performed on the ccurcy of resolving the locl pressure coefficient nd glol integrted force coefficients for smooth susonic nd non-smooth trnsonic flows with moving shock wves on pitching irfoil. For smooth susonic flows, the Fourier TSM method offers significnt ccurcy dvntge over the BDF method for the prediction of oth the locl pressure coefficient nd integrted force coefficients. For trnsonic flows where the motion of the discontinuous shock wve contriutes significnt higherorder hrmonic contents to the locl pressure fluctutions, sufficient numer of modes must e included efore the Fourier TSM provides n dvntge over the BDF method. The Fourier TSM, however, still offers etter ccurcy thn the BDF method for integrted force coefficients even for trnsonic flows. A prolem of non-symmetric solutions for symmetrieriodic flows due to the use of odd numers of intervls is uncovered nd nlyzed. A frequency-serching method is proposed for prolems where the frequency is not known priori. The method is tested on the vortex shedding prolem of the flow over circulr cylinder. B Feng Liu fliu@uci.edu School of Aeronutics, Northwestern Polytechnicl University, Xi n 77, Chin Deprtment of Mechnicl nd Aerospce Engineering, University of Cliforni, Irvine, CA 9697, USA Keywords Fourier time spectrl method (TSM) Pitching irfoil Trnsonic flow Non-symmetric solution Computtionl efficiency Vortex shedding flow Frequency serch Introduction Time dependent clcultions re needed for vrious importnt pplictions, such s the study of the internl flow of turomchinery, flutter, nd jet noise [ 5]. Time ccurcy is lso needed for lrge eddy simultion of turulencegenerted noise [6,7] nd prticle-turulence interctions [8,9]. In ddition to improvements in computer hrdwre, improvements in numericl methods lso ply significnt role in reducing the computtionl cost of time dependent clcultions. A dul-time stepping method tht ws originlly developed y Jmeson [] hs een successful for unstedy flow computtions using oth the Euler nd the Nvier Stokes equtions []. Although locl pseudo-time stepping, residul smoothing, nd multigrid method cn e used to ccelerte convergence, the computtionl efficiency of this method is limited. The ckwrd difference formul (BDF) method for the time discretiztion only provides time ccurcy of lgeric orders, specificlly nd-order for the usul fully implicit three-level BDF scheme. Thus, smll rel time steps re needed due to the time ccurcy requirement. For periodic flows, n lterntive method for rel time discretiztion is to represent flow vriles with Fourier series. One of the erliest methods in this ctegory is linerized method proposed y Hll nd Crwley []. In this method ech flow vrile is ssumed to e sum of the timeverged vlue nd smll perturtion. Bsed on the smll perturtion ssumption, the flow governing eqution cn e simplified nd seprted into two sets of equtions. One is 3

2 L. Zhn et l. for the time-verged vriles, nd the other is for the smll perturtion components. Coefficients in the ltter cn e otined y solving the former. Then the smll perturtion components re ssumed to e hrmonic functions with the known frequency. To develop n efficient method tht cn ccount for nonliner effects, Ning nd He [3] proposed nonliner hrmonic method. In this method, ech flow vrile is still expressed s sum of the time-verged vlue nd hrmonic perturtion. However, when the nonliner flow equtions re time-verged, extr unstedy stress terms re generted in the time-verged equtions. These extr terms re computed from the solution of the unstedy perturtion equtions, while the coefficients of the perturtion equtions re determined y the solution of the time-verged equtions. The two sets of equtions re solved simultneously in strongly coupled mnner. Numericl results show tht nonliner effects, such s oscilltion of shock wves, cn e efficiently modeled y the nonliner hrmonic method. The nonliner hrmonic method ws mong the erliest methods tht ccount for nonliner effect y reserving extr unstedy stress terms in the time-verged equtions. However, the unstediness is still evluted y the first hrmonic mode only. Hrmonic modes with higher frequencies re not involved. Therefore, the effectiveness of this method is limited for lrge nd complex temporl vritions. In ddition, the formultion of the method requires lot of effort, especilly for three-dimensionl viscous unstedy flow prolems. To develop method tht includes more hrmonic modes nd cn e esily pplied in engineering prctice, Hll et l. [] proposed the hrmonic lnce method sed on expnding flow vriles with Fourier series. In this method the rel time derivtive term is pproximted y the time spectrl opertor. Fst Fourier trnsform (FFT) nd its inverse counterprt re used in the time spectrl opertor. The resulting equtions on ll instnts re solved simultneously in time domin nd they re coupled only through the time spectrl opertor. Since the pproximted rel time derivtive term reduces to source term, pseudo-time mrching method for stedy-stte clcultion cn e employed. Convergence ccelerting techniques, such s locl time step nd the multigrid method, cn e pplied during the pseudo-time mrching process without ffecting time ccurcy. The hrmonic lnce method hs een successfully pplied in solving two-dimensionl viscous flow pst cscdes. Computtionl results show tht the method is efficient. Strong nonliner effects, such s oscilltion of shock wves cn e ccurtely predicted if the time resolution is sufficiently high. Gopinth et l. [5] pplied the hrmonic lnce method to the computtion of three-dimensionl unstedy flow inside multi-stge turomchinery. In this computtion only one lde pssge is used for ech lde row. The use of the hrmonic lnce method gretly reduced the overll cost of the simultion. The dominnt unstediness nd the min flow fetures, such s the interction etween two lde rows, re cptured efficiently. In their study, Ekici et l. [6] completed the computtions of three-dimensionl periodic inviscid flows round helicopter rotor using the hrmonic lnce method. The method ws reported to e le to resolve significnt unstedy flow fetures with low computtionl cost. Since the equtions resulting from the hrmonic lnce method re well-suited for djoint sensitivity nlysis, rotor optimiztion including improvement of erodynmierformnce nd noise reduction cn e conveniently conducted. There were lso other wys to clculte periodic flows sed on the Fourier expnsion of flow vriles. A nonliner frequency domin solver ws proposed y McMullen et l. [7], which is equivlent to the hrmonic lnce method. With this solver the governing equtions re solved in the frequency domin rther thn the time domin. However, the residul is still clculted in the time domin for simplicity. Trnsformtions etween the time domin nd the frequency domin for oth solutions nd residuls re required. This method hs een shown to e efficient in solving trnsonic flows pst pitching irfoil nd the vortex shedding flow ehind circulr cylinder t rest. A pper y Gopinth nd Jmeson [8] derived the explicit formul of the pproximted rel time derivtive using the ide of hrmonic lnce method. Since discrete Fourier trnsformtion nd its inverse counterprt re not required in computtion, the resulting equtions re solved completely in the time domin. For this reson, this vrint of the hrmonic lnce method is clled the time spectrl method (TSM). We cll this method the Fourier TSM since it is sed on the Fourier expnsion of flow vriles. The sme work y Gopinth nd Jmeson [8] lso pointed out tht stility prolem my e cused y the use of even numers of intervls in period. Thus, using odd numers of intervls seems to e fvorle. However, we found tht non-symmetric solutions re produced for symmetrirolems if period is split into odd numers of intervls. This prolem hs not een reported efore. We nlyze the reson for this prolem, nd sed on tht, we propose the requirements to ensure symmetric solutions. The study y McMullen nd Jmeson [9] estimted the computtionl efficiency of the nonliner frequency domin solver, which is equivlent to the Fourier TSM. The nonliner frequency domin method is demonstrted to e much more efficient thn the nd-order BDF method. However, the evlution is limited to nlyzing the error of the computed integrtion quntities of the surfce pressure coefficient, such s the lift coefficient nd the moment coefficient. To more comprehensively evlute the efficiency dvntge of the Fourier TSM over the nd-order BDF method, we here conduct systemtic error nlysis on the computed temporl vrition of the surfce pressure coefficient itself. 3

3 Fourier time spectrl method for susonic nd trnsonic flows For the periodic flows where the frequency is not known priori, such s vortex shedding flows, grdient-sed method hs een used successfully to serch for the frequency [7,8]. However, the initil guess of the frequency must e close to the correct vlue since the method is loclly optiml. A new method tht updtes the frequency sed on Fourier nlysis of the lift coefficient efore using the grdient-sed method is developed. So the initil guesses of the frequency tht re fr wy from the correct vlue cn e used. In the following sections, firstly, the Fourier TSM is derived nd vlidted with trnsonic flow over the pitching NACA irfoil. For symmetric flow prolems, the nonsymmetric solutions produced y the use of odd numers of intervls re discussed. Then error nlysis on the surfce pressure coefficient nd its integrtion quntities using oth the Fourier TSM nd the nd-order BDF is conducted for trnsonic flows s well s susonic flows. Finlly, the new frequency serch pproch is proposed nd pplied in conjunction with the Fourier TSM to vortex shedding flow ehind circulr cylinder t rest. The Fourier time spectrl method (TSM) The Nvier Stokes equtions for two-dimensionl compressile viscous flow cn e written s w t where + f c x + g c y f v x g v y ρ w = ρu ρv, f c = ρ E f v = τ xx τ yx uτ xx + vτ yx q x ρū ρuū+ p ρvū ρ Eū + pu, g v = =, (), g c = ρ v ρu v ρv v + p ρ E v + pv τ xy τ yy uτ xy + vτ yy q y, (). In the preceding equtions, t is time, x nd y re position coordintes, ρ is density, p is pressure, (u,v) is the locl flow velocity. The totl energy is E = e + (u + v ) with p internl energy e = (γ )ρ. γ is the rtio of specific het. (ū, v) = (u u,v v ) stnds for the locl convective velocity reltive to the control surfce moving t the velocity of (u,v ). The components of the viscous stress tensor nd those of the het flux vector re defined s elow (3) [ u τ xx = μ x ( u 3 x + v y ( u τ yx = τ xy = μ y + v ), x [ v τ yy = μ y ( u 3 x + v y q x = μ h Pr x, q y = μ h Pr y, )], )], where μ is the moleculr viscosity, which is clculted y Sutherlnd s lw. Pr is the Prndtl numer, h = e + p ρ is enthlpy. If the unstedy flow is periodic, then the flow solution vector w cn e expnded using the Fourier series s elow w(t) = + () w k e ikωt, (5) where ω is the fundmentl ngulr frequency of the periodic flow. The Fourier coefficient w k cn e clculted s w k = ω π ω π w(t)e ikωt dt, k =, ±, ±,... (6) If finite hrmonic modes re retined in the series of Eq. (5), discrete Fourier trnsform (DFT) pir cn e constructed. Assume period is eqully divided into N intervls, then the solution vectors on the left N instnts form the following time sequence, which is the extended solution vector w =[w,...,w n,...,w N ] T. (7) The DFT on the ove extended solution vector is written s w k = N N n= w n e ikωn t, (8) where t = ωn π.ifn is odd, the inverse DFT (IDFT) is written s w n = N k= N w k e ikωn t. (9) When N is even, the inverse DFT is modified s w n = N k= N w k e ikωn t. () 3

4 L. Zhn et l. In mtrix form, the DFT in Eq. (8) cn e expressed s w = Ew, () where E is the DFT opertor mtrix. w is the Fourier coefficient vector of solution. If N is odd, [ w = w, w,..., w N When N is even, w ecomes ] T., w N,..., w () [ T. w = w, w,..., w N +, w N, w N ],..., w (3) For the IDFT in Eq. (9) oreq.(), it cn e expressed in mtrix form s w = E w, () where E is the IDFT opertor mtrix. w nd w form the DFT pir for flow solution. Since the first-order derivtive of periodic function is lso periodic function, the DFT pir for the time derivtive of flow solution cn e constructed s well. The corresponding DFT nd IDFT re w () = Ew () (5) nd w () = E w (), (6) where the extended vector in time domin w () =[ w t,..., w t n,..., w t N ] T, nd w () is the Fourier coefficient vector for the time derivtive of flow solution. The k-th element of w () cn e clculted from the k-th element of w through the following reltion: w () k = ikω w k. (7) Written in mtrix form, w () cn e formulted s w () = iωn F w, (8) where N F is digonl mtrix. If N is odd, N F = dig(,,..., N, N,...,). When N is even, N F = dig(,,..., N +,, N,...,). In TSM, the extended time derivtive vector w () cn e clculted from w through DFT nd IDFT s elow w () = E w () =iωe N F w =iωe N F Ew = Fw, (9) where F denotes the time spectrl opertor, which is n N N mtrix. If N is even, the DFT nd IDFT cn e replced y FFT nd IFFT so s to evlute the time spectrl opertor efficiently. The element t the l-th row nd the n-th column of the time spectrl opertor mtrix, F l,n ( l, n N ), cn e expressed s N πik N (l n) F l,n = ω ike N k= N + ω π(l n) = ( )(l n) cot[ ], l = n, N, l = n. () Note tht the fundmentl ngulr frequency of periodic flow prolem ω is n explicitly specified prmeter inside the time spectrl opertor. Following the method of lines, the unstedy governing eqution () cn e integrted over grid cells using the cellcentered finite volume method. The governing equtions in the semi-discrete form on N eqully spced instnts cn e collected together nd written s dw + R(w ) =. () dt Using the Fourier TSM to solve the ove eqution, the time derivtive term of the eqution cn e pproximted y the time spectrl opertor. The resultnt eqution ecomes Fw + R(w ) =. () The ove eqution tkes the form of governing eqution for stedy prolems. There is no explicit time derivtive nd the time derivtive term reduces to source term. To solve such system, pseudo-time mrching technique is usully dopted. Introducing the pseudo-time τ, the finl eqution to e solved is dw dτ + Fw + R(w ) =. (3) In Eq. (3), R(w ) is clculted y the Jmeson-Schmidt- Turkel (JST) scheme []. The pseudo-time mrching process is conducted y five-stge explicit Runge Kutt scheme ecuse of its extended stility rnge. The fully discretized governing equtions on ll instnts re coupled only through the time spectrl opertor nd re solved simultneously in time domin s in Ref. []. Convergence ccelertion technique, such s locl time stepping nd multigrid method, cn e used during pseudo-time mrching without losing time ccurcy. 3

5 Fourier time spectrl method for susonic nd trnsonic flows 3 Appliction to the pitching irfoil test cse The periodic flows over pitching NACA irfoil t trnsonic flow conditions re frequently used s test cses for unstedy solvers. For this reson, the Fourier TSM is vlidted y solving n inviscid test cse of this kind. Computtionl results using the Fourier TSM is lso compred with those using the BDF method. In the present inviscid test cse, NACA irfoil is forced to pitch round its qurter chord t M =.755. The forced pitching movement is given y α = α + α m sin ωt, () where α is the men ngle of ttck, α m the pitching mplitude. ω is the ngulr frequency nd is relted to the reduced frequency κ s κ = ωc U, (5) where c is the chord length nd U is the free-strem velocity. In this cse, α =.6, α m =.5, nd κ =.8. The periodic inviscid flow in this test cse is clculted on 6 33 O-type grid. The convergence criteri is set to e. To implement the Fourier TSM, three different time resolutions re used. One period is eqully split into 8, 6, or 3 intervls, respectively. For the BDF method, 3 or 6 equl intervls re used. The computed lift nd moment coefficient vritions with respect to the ngle of ttck re shown in Fig.. The experimentl dt t Reynold s numer Re = []relso shown to demonstrte the sme generl greement etween the computtions nd the experiment s found in Go et l. [,3], despite the noticele differences due to negligence of viscous nd turulence effects y the Euler equtions. The purpose of this pper is to investigte the time spectrl method vs. the conventionl BDF time-mrching method in resolving time evolution for the Euler equtions. Therefore, the computtionl result using the BDF method with 6 time intervls is used s the enchmrk ccurte solution in the following discussions. Figure shows tht for the lift coefficient, the computtionl result using the Fourier TSM only with eight intervls grees very well with the ccurte solution. For the moment coefficient, Fig. shows tht the computtionl results using the Fourier TSM converge fst to the ccurte solution s time resolution increses. However, to predict well ll detils of the temporl vrition of the moment coefficient, t lest 6 intervls re needed for the Fourier TSM. Compred to the temporl vrition of the lift coefficient, tht of the moment coefficient is more complicted. Hence, to rech the sme level of ccurcy, more intervls (more Fourier modes in the frequency domin) re required for the Fourier TSM to clculte the ltter. As for the temporl vrition of the surfce pressure coefficient distriution, it is convenient to show it in frequency domin. The time-verged component nd the first three modes re shown in Figs., 3,, nd 5. Comput-, time-verged Frction of chord Fourier TSM, 3 intervls Fourier TSM, 6 intervls Fourier TSM, 8 intervls BDF, 6 intervls BDF, 3 intervls Fig. Inviscid flow over pitching NACA irfoil: time-verged surfce pressure coefficient. Fourier TSM, 3 intervls Fourier TSM, 6 intervls Fourier TSM, 8 intervls BDF, 6 intervls BDF, 3 intervls.5 Fourier TSM, 3 intervls Fourier TSM, 6 intervls Fourier TSM, 8 intervls BDF, 6 intervls BDF, 3 intervls Lift coefficient c l. -. Moment coefficient c m Angle of Attck ( ) Angle of Attck ( ) Fig. Inviscid flow over pitching NACA irfoil: lift nd moment coefficients versus ngle of ttck. Lift coefficient. Moment coefficient 3

6 L. Zhn et l., first mode, rel component. -. Fourier TSM, 3 intervls Fourier TSM, 6 intervls Fourier TSM, 8 intervls BDF, 6 intervls BDF, 3 intervls, first mode, imginry component.. -. Fourier TSM, 3 intervls Fourier TSM, 6 intervls Fourier TSM, 8 intervls BDF, 6 intervls BDF, 3 intervls Frction of chord Frction of chord Fig. 3 Inviscid flow over pitching NACA irfoil: first mode of surfce pressure coefficient. Rel component. Imginry component, second mode, rel component Fourier TSM, 3 intervls Fourier TSM, 6 intervls Fourier TSM, 8 intervls BDF, 6 intervls BDF, 3 intervls, second mode, imginry component Fourier TSM, 3 intervls Fourier TSM, 6 intervls Fourier TSM, 8 intervls BDF, 6 intervls BDF, 3 intervls Frction of chord Frction of chord Fig. Inviscid flow over pitching NACA irfoil: second mode of surfce pressure coefficient. Rel component. Imginry component, third mode, rel component Fourier TSM, 3 intervls Fourier TSM, 6 intervls Fourier TSM, 8 intervls BDF, 6 intervls BDF, 3 intervls, third mode, imginry component Fourier TSM, 3 intervls Fourier TSM, 6 intervls Fourier TSM, 8 intervls BDF, 6 intervls BDF, 3 intervls Frction of chord Frction of chord Fig. 5 Inviscid flow over pitching NACA irfoil: third mode of surfce pressure coefficient. Rel component. Imginry component tionl results s well s experimentl dt [] ll revel tht the rnge of the shock wve movement is pproximtely. < x/c <.6. Within this re, the Fourier TSM with eight intervls in period cn only resonly predict the time-verged component of the surfce pressure coefficient. When time resolution increses to hve 6 intervls in period, the Fourier TSM could lso generlly resolve the first mode, the second mode nd the rel component of the third mode. For the imginry component of the third mode, Fig. 5 shows tht the computtionl result using the Fourier TSM with 6 intervls devites drmticlly from the ccurte solution. To mke improvement on this, 3 intervls re needed. Outside the shock wve movement re (x/c <. orx/c >.6), the Fourier TSM with 3

7 Fourier time spectrl method for susonic nd trnsonic flows only eight intervls could perfectly resolve the time-verged component nd ll three modes of the surfce pressure coefficient. This is ecuse in the shock-free re, the temporl vrition of the surfce pressure coefficient is smooth, its Fourier modes with high frequencies decy rpidly. Thus, only retining the first three modes(corresponding to eight intervls) is sufficient for the Fourier TSM to otin ccurte computtionl results. Since the temporl vrition of the surfce pressure coefficient could e perfectly predicted in most re, the Fourier TSM with only eight intervls is le to clculte ccurtely the integrtion quntities, in prticulr the lift coefficient. For the moment coefficient, the ccurcy loss of the surfce pressure coefficient in the re of shock wve movement only ffects some detils of its temporl vrition. The generl trend nd the rnge of the temporl vrition re out correct., time-verged Frction of chord Fourier TSM, 7 intervls Fourier TSM, 8 intervls BDF, 3 intervls Fig. 6 Symmetric inviscid flow over pitching NACA irfoil: timeverged surfce pressure coefficient Non-symmetric solutions of symmetrirolems due to odd numers of intervls When the Fourier TSM is pplied to solve periodic flow prolems, proper time resolution should e crefully selected to lnce ccurcy of the solution nd computtionl cost. Another importnt issue is whether the numer of intervls is even or odd. Gopinth nd Jmeson [8] pointed out tht the use of n even numer of intervls my cuse stility prolem for the Fourier TSM, prticulrly in the cses where the time derivtive is significnt, such s high RPM turomchinery prolems. This is ecuse the odd-even decoupled solution my e llowed when the Fourier TSM is pplied with n even numer of intervls. Thus, using odd numers of intervls seems to e fvorle. However, we found tht the Fourier TSM produces non-symmetric solutions for symmetric flow prolems if n odd numer of intervls re employed. To illustrte this, symmetric flow prolem is constructed from the previous test cse nd is solved y the Fourier TSM. In this symmetric flow prolem, the irfoil still pitches round its qurter chord point. All the flow conditions remin the sme except tht the men ngle of ttck is set to e zero. Under these flow conditions, the periodic flow pst the irfoil is symmetric given tht the irfoil is symmetric. For this symmetric flow, the distriution of the surfce pressure coefficient on the upper surfce should e perfectly symmetric to or overlp tht on the lower surfce in frequency domin. The computed time-verged component nd the first three Fourier modes re shown in Figs. 6, 7, 8, nd 9, respectively. For the Fourier TSM, seven or eight intervls re used. The solution otined y the BDF method with 3 intervls serves s reference solution. It is oserved tht when eight intervls re used, the solution from the Fourier TSM is symmetric, just like the reference solution. This is true for ny Fourier mode of the pressure coefficient in ny surfce re. Wheres, if seven intervls re employed, non-symmetric solution is produced y the Fourier TSM, especilly in the re where shock wve moves. Outside the re of shock wve movement, the solution is still non-symmetric nd the symmetry ecomes quite wek. Whether the symmetry of the solution using seven intervls in certin re is ovious or not ctully depends on if seven intervls re sufficient to resolve the locl unstedy flow. The computtionl results of the previous test cse for vlidtion hve shown tht using eight intervls could only retin three Fourier modes in the Fourier TSM. This is not sufficient to resolve the shock wve movement. Thus, the clculted surfce pressure coefficient exhiits drmtic error in the re where shock wve moves. Using seven intervls cn only retin the first three Fourier modes in the Fourier TSM s well. Thus, the error of the computed surfce pressure coefficient is lso high in the sme surfce re. This is why the symmetry of the solution using seven intervls is ovious. It cn e nticipted tht if time resolution is incresed so s to retin more Fourier modes in the Fourier TSM, the symmetry of solutions using odd numers of intervls will decrese in the re of shock wve movement. Actully, the prolem of non-symmetric solution of symmetric prolem due to the use of odd numers of intervls is not limited to the Fourier TSM. Numericl experiments hve confirmed tht the sme prolem lso hppens to the BDF method. The reson for this prolem is not ssocited with the specific method of time discretiztion, ut lies in the wy independent instnts (not including the one t the end of period) re distriuted in period. To gurntee symmetric solutions for symmetric flow prolems, we propose tht the distriution of independent instnts should stisfy the following requirements. All independent instnts should e le to e grouped into pirs, nd the phse difference etween 3

8 L. Zhn et l.., first mode, rel component. Fourier TSM, 7 intervls Fourier TSM, 8 intervls BDF, 3 intervls , first mode, imginry component Fourier TSM, 7 intervls Fourier TSM, 8 intervls BDF, 3 intervls Frction of chord Frction of chord Fig. 7 Symmetric inviscid flow over pitching NACA irfoil: first mode of surfce pressure coefficient. Rel component. Imginry component, second mode, rel component TSM (7 intervls) TSM (8 intervls) BDF (3 intervls) Frction of chord, second mode, imginry component TSM (7 intervls) TSM (8 intervls) BDF (3 intervls) Frction of chord Fig. 8 Symmetric inviscid flow over pitching NACA irfoil: second mode of surfce pressure coefficient. Rel component. Imginry component, third mode, rel component. TSM (7 intervls) TSM (8 intervls) BDF (3 intervls) Frction of chord, third mode, imginry component TSM (7 intervls) TSM (8 intervls) BDF (3 intervls) Frction of chord Fig. 9 Symmetric inviscid flow over pitching NACA irfoil: third mode of surfce pressure coefficient. Rel component. Imginry component ech pir of independent instnts should e 8. Apprently, splitting period into eight equl intervls (or other even numers of intervls) utomticlly stisfies ll these requirements. Hence, the solution for symmetrirolem is symmetric. If period is divided into seven equl intervls (or other odd numers of intervls), it is impossile to group ll the independent instnts into pirs since the numer of independent instnts is odd. Thus, the requirements 3

9 Fourier time spectrl method for susonic nd trnsonic flows re not stisfied nd the solution for symmetrirolem is non-symmetric. For the reson discussed ove, we choose to use even numers of equl intervls in period for the Fourier TSM s long s the computtion cn converge. Gopinth lso justified tht the Fourier TSM using n even numer of intervls is usully stle for prolems where the time derivtive is reltively smll, such s the flow pst pitching irfoils nd wings t low forced frequencies [8]. In ny of the present computtions using the Fourier TSM with even numers of intervls, the stility prolem is not encountered. A possile reson is tht the frequencies in these flow prolems re not high enough to trigger the stility prolem. In ddition to using even numers of intervls, we lso mke sure tht the numer of intervls is power of since efficient FFT cn e directly used in the Fourier TSM. 5 Error nlysis on solutions for trnsonic nd susonic flows For smooth flows, time resolution y the Fourier TSM should e of exponentil order of ccurcy s compred to the ndorder time ccurcy of the BDF scheme used in Ref. []. However, this my not e the cse for flows with shock motion. To evlute comprehensively the computtionl efficiency of the Fourier TSM nd compre it to tht of the BDF method, error nlysis is conducted to the computed solutions. The previous test cse for vlidtion hs shown tht whether or not shock wve movement exists hs gret influence on the time ccurcy of the Fourier TSM. For this reson, two flow prolems re solved to provide solutions for error nlysis. The first one is still the previous test cse for vlidtion, in which the NACA irfoil pitches round its qurter chord point with M =.755, α =.6, α m =.5, nd κ =.8. In this flow prolem, shock wves move ck nd forth over irfoil surfce. For the purpose of convenient comprison, the second flow prolem is constructed from the first one. In the constructed flow prolem, ll the flow conditions remin the sme except tht the free strem Mch numer is lowered to M =.6. Computtionl results show tht the flow of this prolem is susonic everywhere. Hence, this flow prolem is shock-free. To conduct error nlysis, five different time resolutions re used for the Fourier TSM. Corresponding to these time resolutions, period is split into, 8, 6, 3, or 6 equl intervls. When the BDF method is pplied, 6, 3, or 6 equl intervls re used in period. The solution using the Fourier TSM with 6 intervls is selected s the ccurte solution. To ensure the reliility of the error nlysis, computtions using the Fourier TSM nd the BDF method ll converge to the residul level of. McMullen nd Jmeson [9] chose to use lift nd moment coefficients s figures-of-merit when evluting the computtionl efficiency of the Fourier TSM. As the first step of the present error nlysis, we follow the sme ide to work on the computed lift nd moment coefficients s well. For ny computed solution, the squred error of the lift coefficient in period is defined s follows Error c l = m n= [ (Re n (c l ) Re n (c l, )) + (Im n (c l ) Im n (c l, )) ] + m n=m+ [ (Re n (c l, )) + (Im n (c l, )) ], (6) where Re n (c l ) nd Im n (c l ) re the rel nd imginry components of the n-th Fourier mode of the lift coefficient for the given solution. Re n (c l, ) nd Im n (c l, ) re the rel nd imginry components of the n-th Fourier mode of the lift coefficient for the ccurte solution. m nd m re the numers of Fourier modes tht the given solution nd the ccurte solution include, respectively. The first summtion in the Eq. (6) represents the lising error of the given solution, wheres the second one is mesure of the trunction error of the sme solution. Similrly, for ny computed solution, the squred error of the moment coefficient in period cn e defined s follows Error c m = m n= [ (Re n (c m ) Re n (c m, )) + (Im n (c m ) Im n (c m, )) ] + m n=m+ [ (Re n (c m, )) + (Im n (c m, )) ], (7) where Re n (c m ) nd Im n (c m ) re the rel nd imginry components of the n-th Fourier mode of the moment coefficient for the given solution. Re n (c m, ) nd Im n (c m, ) re the rel nd imginry components of the n-th Fourier mode of the moment coefficient for the ccurte solution. According to the definitions in Eqs. (6) nd (7), the results of error nlysis for the lift nd moment coefficients re shown in Fig.. Figure shows tht for the flow with shock wve, the error in the lift coefficient using the BDF method decreses s the time resolution increses, nd the drop rte of the error is less thn. In the log-log coordinte system, if the temporl vrition of the locl pressure coefficient is smooth over the entire irfoil surfce, then the drop rte of its error nd the error of its integrtion quntities(such s lift coefficient) for the nd-order BDF method should e exctly. If shock 3

10 L. Zhn et l. Error in lift coefficient (c l ) Fourier TSM, with shock wve nd-order BDF, with shock wve Fourier TSM, shock free nd-order BDF, shock free Line with slope equl to - Error in moment coefficient (c m ) Fourier TSM, with shock wve nd-order BDF, with shock wve Fourier TSM, shock free nd-order BDF, shock free Line with slope equl to Numer of intervls Numer of intervls Fig. Error of the computed solutions. For lift coefficient. For moment coefficient wve occurs in some region of the irfoil surfce, the temporl vrition of the pressure coefficient is not smooth in the region. Thus, for the solutions of the nd-order BDF method, the drop rte of the error in the pressure coefficient nd tht of the lift coefficient ll decrese elow. Using the sme time resolution, the error of the lift coefficient for the Fourier TSM is lower thn tht for the BDF method. This indictes tht to otin the lift coefficient with the sme ccurcy level, much fewer intervls re needed for the Fourier TSM. For instnce, the lift coefficient clculted y the Fourier TSM with eight intervls is s ccurte s the one otined y the BDF method with 6 intervls. So it cn e concluded tht if the lift coefficient is tken s the figure-of-merit, the Fourier TSM is much more efficient thn the nd-order BDF method even when shock wve occurs. Figure lso shows tht in the shock-free cse, the drop rte of the error in the lift coefficient is exctly for the ndorder BDF method. This is n expected result ccording to theoreticl nlysis. Under the sme time resolution, the error in the lift coefficient for the Fourier TSM is much lower thn tht for the BDF method. For instnce, the lift coefficient clculted y the Fourier TSM with only four intervls is s ccurte s the one otined y the BDF method with 6 intervls. This mens if the lift coefficient is tken s the figure-of-merit, the Fourier TSM is extremely efficient for the shock-free cse. Figure shows tht the conclusion of the error nlysis for the moment coefficient is similr to tht for the lift coefficient. However, if the moment coefficient is tken s the figure-of-merit, the efficiency dvntge of the Fourier TSM over the BDF method is wekened especilly for the flow with shock wve. This is ecuse the temporl vrition of the moment coefficient is much more complicted thn tht of the lift coefficient if shock wve occurs. For this reson, higher time resolution is required for the Fourier TSM to resolve the detils of the temporl vrition of the moment coefficient. The ove error nlysis is mde on the integrl quntities of the surfce pressure coefficient. In prctice, the computtionl result of the surfce pressure itself is lso of gret importnce since it cn reflect locl flow detils nd key fetures. To evlute the computtionl efficiency of the Fourier TSM more comprehensively, the error nlysis should e conducted on the computed surfce pressure coefficient s well. The following verged squred error of the surfce pressure coefficient cn e defined for given solution: Error,verged = q q m s= n= { [Re n ( (s)) Re n (, (s))] +[Im n ( (s)) Im n (, (s))] } + q q m s= n=m+ +[Im n (, (s))] }, { [Re n (, (s))] (8) where Re n ( (s)) nd Im n ( (s)) re the rel nd imginry components of the n-th Fourier mode of the pressure coefficient for the given solution. Re n (, (s)) nd Im n (, (s)) re the rel nd imginry components of the n-th Fourier mode of the pressure coefficient for the ccurte solution. q is the numer of grid cells over the irfoil surfce. According to the definition in Eq. (8), the results of error nlysis for the surfce pressure coefficient is shown in Fig.. The verge error of the surfce pressure coefficient drops s time resolution increses in the similr trend tht cn e oserved for the lift coefficient or the moment coefficient. However, for the flow with shock wve, the ver- 3

11 Fourier time spectrl method for susonic nd trnsonic flows Averge error in pressure coefficient ( ) Fourier TSM, with shock wve nd-order BDF, with shock wve Fourier TSM, shock free nd-order BDF, shock free Line with slope equl to Numer of intervls Fig. Averged error of the surfce pressure coefficient ge error of the surfce pressure coefficient for the Fourier TSM ecomes comprle to tht for the BDF method if the time resolution is reltively low (6 or fewer intervls re used in period). For the flow with shock wves, the spectrum of the surfce pressure coefficient contins significnt components of higher modes due to the motion of the discontinuous shock wve. They only strt to decy drmticlly fter the 7-th Fourier mode (seven modes re included if 6 intervls re used in period). In the shock-free re, the spectrum of the surfce pressure coefficient is much simpler nd ll Fourier modes decy rpidly. When time resolution is reltively low, the verge error of the surfce pressure coefficient minly reflects the locl error in the region of shock wve movement. In tht region if 6 or fewer intervls re used, dominnt Fourier modes re not included completely in the Fourier TSM. Hence, the lising error nd the trunction error re high. Recll tht the lift coefficient hs very simple spectrum nd ll of its Fourier modes decy rpidly. Hence, the temporl vrition of lift coefficient cn e predicted ccurtely y the Fourier TSM even when the time resolution is pretty low. The ove error nlysis demonstrtes tht for integrl quntities of the surfce pressure coefficient (especilly the lift coefficient) in flows with shock wves, the Fourier TSM is much more efficient thn the BDF method. If the temporl vrition of the surfce pressure coefficient itself is simulted, the computtionl efficiency of the Fourier TSM decreses, ut is still not lower thn tht of the BDF method. For the shock-free flow prolems, the Fourier TSM is extremely efficient for oth the surfce pressure coefficient nd its integrl quntities. Given the numer of pseudo-time steps needed to rech convergence t ech time instnce eing out the sme, the totl CPU time required y ech method depends on the numer of time instnces to e computed in period. The Fourier TSM, eing of exponentil order of ccurcy, chieves much higher time ccurcy thn the conventionl nd-order BDF method, even for flows with shocks s discussed ove. As result, to chieve the sme level of time ccurcy, fewer time instnces (intervls) in period re needed for the Fourier TSM thn for the conventionl BDF time-mrching method. For exmple, Fig. shows tht the Fourier TSM with eight time intervls chieves the sme ccurcy s the BDF with 6 time intervls for the shock-free cse. In ddition, to rech stedy-stte periodic solution, the conventionl timemrching solver hs to go through n initil trnsient process y mrching the rel time forwrd for severl (usully five or more) periods. With the Fourier TSM, however, the stedystte periodic solution is otined in one shot y solving the coupled time-spce equtions without hving to go through n extr trnsient period. Comining these two fctors, the Fourier TSM is orders of mgnitude fster thn the conventionl time-mrching method for periodic flow prolems where high-order of ccurcy in time is needed. In prctice, however, the computtionl efficiency dvntge of the Fourier TSM gretly hinges on the efficiency in solving the coupled system of Eq. () nd lso on the smoothness of the solutions. For prolems where the solution contins discontinuities such s shock wves, the dvntge of the Fourier TSM time my e reduced. 6 Appliction to the periodirolems with unknown frequency In the preceding computtions of the flows over the pitching NACA irfoil, the frequency of the flow prolems is known since it is equl to the given pitching frequency. For such computtions, the Fourier TSM cn e directly pplied. Recll tht the frequency is n explicit fctor in the time spectrl opertor nd it must e given for the Fourier TSM. There lso exist flow prolems in which the frequency is not known efore experiments or computtions. A representtive flow prolem of this kind is the lminr vortex shedding flow ehind circulr cylinder. Though the flow is known to e periodic, the frequency is not known priori. Only rough estimtion cn e mde y empiricl formul. To otin the ccurte frequency nd the unstedy flow field using the Fourier TSM or equivlent methods, frequency serching process must e involved. McMullen et l. [7] proposed grdient sed vrile time period (GBVTP) method for the frequency domin method. Gopinth nd Jmeson [8]proposed similr grdient sed method to the Fourier TSM. In the grdient-sed method, the frequency is updted using the negtive grdient of the squred unstedy residul with respect to frequency s follows ω l+ = ω l α R ω, (9) 3

12 L. Zhn et l. where α is coefficient tht controls the updte nd it hs to e crefully chosen to gurntee convergence. Given the initil guess of the frequency tht is sufficiently close to the correct vlue, the method cn find the exct frequency precisely. To roden the serch rnge nd mke the initil guess of frequency less constrined, we propose new method tht is sed on Fourier nlysis of the lift coefficient (or nother locl or glol flow vrile) to estimte the frequency efore the grdient-sed method is used to otin the finl converged vlue. For prolems like vortex shedding flow over circulr cylinder, the first Fourier mode of the lift coefficient usully chieves mximum mplitude. Applying Fourier nlysis to the lift coefficient, the frequency of the Fourier mode with mximum mplitude cn e ssigned to e the new frequency. When the first mode of the lift coefficient reches mximum mplitude, the grdient-sed method tkes over the duty of serching for the correct frequency. In the present computtion, circulr cylinder is fixed in the lminr flow t M =. nd Re = 8. A 57 9 O-type mesh is used. The norml distnce from the first grid point to the surfce is 3, while the cylinder dimeter is unity. The fr-field oundry of the mesh extends to out dimeters wy from the cylinder. Bsed on the experience from the preceding computtions, eight rel-time intervls re employed when implementing the Fourier TSM. Since the present flow prolem is shock-free, eight intervls should e sufficient to pply the Fourier TSM. Convergence criteri for the unstedy flow field is set to e 6. The frequency serch process doesn t stop until the flow field converges. The present computtionl result is compred with experimentl dt [ 7]. The convergence history is shown in Fig.. Using the Fourier TSM nd the proposed frequency serch pproch, the residul of the computtion cn finlly drop to the conver- Tle Vortex shedding flow ehind circulr cylinder Willimson [,5].99 Roshko [6].85 Henderson [7].336 Current computtion St Time-verged c d gence criteri. The frequency updting history with respect to multigrid cycles is shown in Fig.. The non-dimensionl ngulr frequency is defined s ω = ωd p ρ, (3) where d is dimeter of cylinder, p the free-strem pressure, nd ρ the free-strem density. The initil nondimensionl ngulr frequency is out.37 (the corresponding reduced frequency is.5), while the finl converged non-dimensionl ngulr frequency is.85. The initil guess of the frequency is fr wy from the correct vlue. Finlly, the Strouhl numer is found nd the converged time-verged drg coefficient of the present computtion re compred with experimentl dt in Tle. Forthe Strouhl numer, the current computtionl result is very close to Willimson s [,5] dt. For the time-verged drg coefficient, the current computtionl result mtches Henderson s [7] dt very well. In the present computtion, Fourier nlysis of the lift coefficient is crried out fter every multigrid cycles. The histories of lift coefficient t different time levels with respect to multigrid cycles re shown in Figs. 3,, nd 5. Figure 3 shows tht during the first multigrid cycles, Residul Nondimensionl ngulr frequency ω Multigrid cycles 3 Multigrid cycles Fig. Convergence history of the flow over circulr cylinder cse. History of residul. History of computed frequency ω 3

13 Fourier time spectrl method for susonic nd trnsonic flows Lift coefficient c l t= t=t/8 t=t/8 t=3t/8 t=t/8 t=5t/8 t=6t/8 t=7t/8 Amplitude of Fourier mode Multigrid cycles - 3 Fourier mode of c l Fig. 3 Flow over circulr cylinder: lift coefficient nd nd its Fourier spectrum in multigrid cycle. c history in multigrid cycle. Spectrum of c l t the end of multigrid cycle Lift coefficient c l t= t=t/8 t=t/8 t=3t/8 t=t/8 t=5t/8 t=6t/8 t=7t/8 Amplitude of Fourier mode Multigrid cycles - 3 Fourier mode of c l Fig. Flow over circulr cylinder: lift coefficient nd nd its Fourier spectrum in multigrid cycle. c history in multigrid cycle. Spectrum of c l t the end of multigrid cycle.8.6 t= t=t/8 t=t/8 t=3t/8 t=t/8 t=5t/8 t=6t/8 t=7t/8.8.6 t= t=t/8 t=t/8 t=3t/8 t=t/8 t=5t/8 t=6t/8 t=7t/8 Lift coefficient c l.. Lift coefficient c l Multigrid cycles 3 5 Multigrid cycles Fig. 5 Flow over circulr cylinder: lift coefficient in multigrid cycles fter. c history in multigrid cycle 3. c history fter multigrid cycle 3 3

14 L. Zhn et l. the lift coefficient t different time levels generlly oscillte with the sme phse. However, phse difference strts to pper ner the -th multigrid cycle. Figure 3 shows tht t the end of the -th multigrid cycle, the third Fourier mode of the lift coefficient chieves mximum mplitude. Since the third Fourier mode is the one with the highest frequency tht the Fourier TSM with eight intervls cn include, the overll error of this mode is usully the highest mong the included Fourier modes. For this reson, the frequency of the second Fourier mode is ssigned s the new frequency. Figure shows tht in the next multigrid cycles, the lift coefficient on different time levels continues to oscillte with phse difference. However, the mplitude of the oscilltion is growing grdully. Figure shows tht t the end of the -th multigrid cycle, the first Fourier mode of the lift coefficient lredy chieves mximum mplitude. So Drg coefficient c d Drg due to pressure, originl Drg due to pressure, reuilt Drg due to skin friction, originl Drg due to skin friction, reuilt Totl drg, originl Totl drg, reuilt from the -th multigrid cycle, the grdient sed method strts to serch the correct frequency. In the third multigrid cycles, Fig. 5 shows tht the lift coefficient on ech time level stops oscillting grdully nd pproches its converged vlue. After tht, the lift coefficient on ech time level completely stops oscillting nd finlly converged. This is illustrted in Fig. 5. The temporl vritions of the totl drg coefficient nd its components due to pressure nd skin friction in period re shown in Fig. 6. The originl discrete computtionl results, s well s the reuilt results re plotted. The reuilt results re otined y summing the resolved Fourier modes on 9 eqully spced instnts over the otined period. It cn e oserved tht the drg due to pressure is the dominnt component of the totl drg. The men vlue nd the vrition mplitude of the drg due to pressure re ll much lrger thn those of the drg due to skin friction. The temporl vritions of the lift nd moment coefficients in period re shown in Fig. 7. Figure 8 shows the vortex shedding process during one shedding period. Mch numer contours in flood type nd stremlines re plotted using the present computtionl results. Every vortex in this flow is initilly generted on the surfce re ner the triling edge, then it grows lrger progressively. When the vortex is out the size of the cylinder, it detches from the surfce. After tht, the detched vortex sheds wy from the cylinder nd dissiptes in the wke. This process occurs lterntely on upper nd lower surfce re ner the triling edge Frction of period Fig. 6 Flow over circulr cylinder: drg coefficients in period 7 Conclusion The Fourier TSM for the unstedy Nvier Stokes equtions is presented nd tested for trnsonic flow over pitching.8.6 Originl Reuilt.8 Originl Reuilt.6 Lift coefficient c l Moment coefficient c m Frction of period Frction of period Fig. 7 Flow over circulr cylinder: lift nd moment coefficients in period. Lift coefficient. Moment coefficient 3

15 Fourier time spectrl method for susonic nd trnsonic flows 3 3 Y Y M: M: X c d M: M: X X 6-8 h X M: Y Y 8-3 M: Y Y f - X -3 g 8 - e 6 3 Y Y - X -3 M: X M: X 6 8 Fig. 8 Flow over circulr cylinder: mch numer contours nd strem lines during one vortex shedding period. t =. t = d t = 38 T. e t = 8 T. f t = 58 T. g t = 68 T. h t = 78 T 8T. ct = 8T. 3

16 L. Zhn et l. NACA irfoil nd compred with the results of using conventionl nd-order BDF method. Convergence study of the time discretiztion errors with respect to the numer of time steps for the two methods show tht the Fourier TSM offers significntly etter ccurcy thn the secondorder BDF method for predicting oth the locl pressure coefficient nd integrted force coefficients for susonieriodic flows. For trnsonieriodic flows where the motion of the discontinuous shock wve contriutes significnt higherorder hrmonic components to the locl pressure, sufficient numer of modes must e included efore the Fourier TSM provides n dvntge over the BDF method. For exmple, more thn 6 time intervls re needed in period for the Fourier TSM to resolve the surfce pressure coefficient in regions of the shock-wve movement for the oscillting irfoil prolem. The Fourier TSM, however, still offers etter ccurcy thn the BDF method for integrted force coefficients even for flows with shock wves. Computtions lso revel prolem of non-symmetric solutions occurring for symmetrieriodic flows due to the use of odd numers of intervls. This non-symmetry is ccentuted for the Fourier TSM for trnsonic flows when n insufficient numer of hrmonic modes re included. For prolems where the frequency is not known priori, serch lgorithm sed on comintion of Fourier nlysis of the computed time-history of flow quntity nd grdient method sed on minimizing the unstedy residul of the Nvier Stokes equtions cn e used to quickly nd ccurtely determine the frequency of the flow. This frequency-serch method long with the Fourier TSM re successfully demonstrted for the periodic vortex shedding prolem of the low Reynolds numer flow over circulr cylinder. Fourier nlysis of the lift coefficient is first performed to estimte the unknown frequency efore the grdient method is used to pin down the finl vlue. The method works for initil guesses of the frequency tht re fr wy from the correct vlue. The Fourier TSM gives excellent prediction of the drg vlue nd the Strouhl numer compred to experimentl dt with s few s only eight time intervls for this vortex shedding flow prolem. Acknowledgments The reserch ws supported y the Stte Scholrship Fund of the Chin Scholrship Council (Grnt 9699). References. Yo, J., Jmeson, A., Alonso, J.J., et l.: Development nd vlidtion of mssively prllel flow solver for turomchinery flows. J. Propuls. Power 7, (). Sdeghi, M., Liu, F.: Inverstigtion of mistuning effects on cscde flutter using coupled method. J. Propuls. Power 3, 66 7 (7) 3. Liu, F., Ci, J., Zhu, Y., et l.: Clcultion of wing flutter y coupled fluid-structure method. J. Aircr. 38, 33 3 (). Xio, Q., Tsi, H.M., Liu, F.: Numericl study of trnsonic uffet on supercriticl irfoil. AIAA J., 6 68 (6) 5. Xiong, J., Nielsen, P., Liu, F., et l.: Computtion of high-speed coxil jets with fn flow deflection. 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