• <tr id="yyy80"></tr>
  • <sup id="yyy80"></sup>
  • <tfoot id="yyy80"><noscript id="yyy80"></noscript></tfoot>
  • 99热精品在线国产_美女午夜性视频免费_国产精品国产高清国产av_av欧美777_自拍偷自拍亚洲精品老妇_亚洲熟女精品中文字幕_www日本黄色视频网_国产精品野战在线观看 ?

    Local and biglobal linear stability analysisof parallel shear flows

    2017-03-13 05:47:08SanjayMittalandAnubhavDwivedi

    Sanjay Mittal and Anubhav Dwivedi

    1 Introduction

    Thehydrodynamic stability of laminar flows has received significant attention and has been investigated by several researchers in the past[Schmid and Henningson(2001);Chandrasekhar(1981);Huerre and Monkewitz(1990);Huerre(2000);Chomaz(2005)].The linear stability of parallel shear flows can be analyzed via finding solution to the Orr-Sommerfeld (OS) equation [Orr (1907); Sommerfeld(1908)], with suitable boundary conditions. The disturbance fi eld is assumed to be a plane wave whose amplitude varies transverse to the flow and is periodic in the homogeneous directions. The analysis can be carried out in either a spatial or temporal framework [Boiko, Dovgal, Grek, and Kozlov (2012)]. The spatial analysis assumes that the disturbance field develops in s pace. The spatial growth rate is determined for different values of frequency and Reynolds number. In contrast, the temporal analysis assumes that the disturbance develops in time. As per the Squire’s theorem [Schmid and Henningson (2001)], the 2D disturbance is the most critical in terms of its growth rate. Therefore, it is suffi cient to consider twodimensional disturbances that have streamwise periodicity [Boiko, Dovgal, Grek,and Kozlov (2012)]. The analysis is carried out to determine temporal growth rate at various Re and for disturbances with different values of streamwise wavenumber. The spatial and temporal approaches for local analysis are related to each other[Huerre (2000)]. For example, Gaster (1962) proposed a transformation for that,approximately, relates the temporal and spatial growth. Several methods have been used to solve the OS equations. Davey and Drazin (1969) utilized Bessel functions to represent the disturbance fi eld and analyze the stability of pipe Poiseuille flow. Orszag (1971) used Chebyshev polynomials to solve the OS equation for the plane Poiseuille flow. Saraph, Vasudeva, and Panikar (1979) used Galerkin’s weighted residual method to carry out the stability analysis of plane Poiseuille flow and magneto-hydrodynamic flows. Garg and Rouleau (1972) used asymptotic analysis to carry out the linear stability analysis in pipe flow. The method has also been applied, in a local sense, to spatially developing flows [Pierrehumbert (1985); Yang and Zebib (1989); Monkewitz (1988); Chomaz, Huerre, and Redekopp (1988)]. In this approach, the flow profi les at different streamwise stations are analyzed by assuming that each profi le corresponds to an independent parallel flow. The local analysis, at each streamwise station of the flow, involves solving the OS equation,with suitable boundary conditions.

    Analternateapproach toinvestigatethelinear stability of fluid flowsisthe BiGlobal and TriGlobal stability analysis[Theofilis(2011);Swaminathan,Sahu,Sameen,and Govindrajan(2011)].Unlike in the local analysis,in this approach the disturbance fi eld is represented globally,including in the streamwise direction.The analysis results in global modes which,depending on the sign of the growth rate,may either grow or decay in the entire computational domain with time.The global analysisisusually muchmorecomputationally expensivethan thelocal one.Such an approach has been used to analyze the global linear stability properties of several non-parallel flows[Mittal(2004);Chomaz(2005);Schmid and Henningson(2001)].Swaminathan,Sahu,Sameen,and Govindrajan(2011)carried out a global linear stability analysis of a diverging channel flow using spectral collocation method.Mittal and Kumar(2003)used astabilized finite element method for the global LSA of stationary and rotating cylinder.Later,Verma and Mittal(2011)used asimilar approachfor carryingout global LSA to investigatetheexistenceand stability of secondary wake mode of a two-dimensional flow past a circular cylinder.Morerecently,Navrose,Meena,and Mittal(2015)carried out LSA of spinning cylinder in auniform flow and identifi ed several unstablethree-dimensional modes for variousrotation ratesof thespinning cylinder.

    In the present work,Linear Stability Analysis(LSA)of the plane Poiseuille flow is carried out.Local and global analyses are considered.The solutions to the OS equation for local analysis have been obtained in a temporal framework.A spectral collocation method based on Chebyshev polynomials[Schmid and Henningson(2001)]is used to solve the governing Orr-Sommerfeld(OS)equation.The global LSA of theplane Poiseuilleflow iscarried out using astabilized finiteelement formulation.The governing equationsarecast in theprimitivevariables:velocity and pressure.Equal-order finite-element interpolation functions are used for pressure and velocity disturbancefi elds.Four-noded quadrilateral elementswith bilinear interpolation isemployed.Thestreamline-upwind/Petrov-Galerkin(SUPG)[Brooks and Hughes(1982)]and pressure-stabilizing/Petrov-Galerkin(PSPG)stabilization techniques[Tezduyar,Mittal,Ray,and Shih(1992)]are employed to stabilize the computations against spurious numerical oscillations.The fi nite element formulation results in a generalized eigenvalue-vector problem which is solved using the subspace iteration method[Stewart(1975)].For carrying out the global analysis,we assume periodic boundary conditions at the inflow and the outflow for the disturbancefield.Thisallowsadirectcomparisonof theglobal LSA withthe OSequation.A comparison between the local and global analysis of the plane Poiseuille flow at Re=7000 is presented and is utilized to show the connection between the two analyses.

    2 Governing Equations

    2.1 Linearized Disturbance Equations

    Let,??Rnsdand(0,T)be the spatial and temporal domains respectively,where nsdis the number of space dimensions,and letΓ denote the boundary of?.The Navier-Stokesequationsgoverning incompressiblefluid flow are given as:

    Hereρ,u andσ are the density,velocity and the stress tensor,respectively.The stresstensor isrepresented asσ =?p I+μ((?u)+(?u)T),where p andμ arethe pressure and coeffi cient of dynamic viscosity,respectively.The boundary conditionsarespecified as:

    Here,ΓgandΓhare the complementary subsetsof the boundaryΓwhere Dirichlet and Neumann boundary conditionsarespecified,respectively.

    To understand the evolution of small disturbances,the unsteady solution is expressed asacombination of steady solution and disturbance:

    Here,U and P representthesteady statesolution whosestability isto bedetermined while u′and p′aretheperturbation fields.Substituting thedecomposition given by Eq.(3)in Eqs.(1)and subtracting from them,the equations for steady flow one obtains the evolution equations for the disturbance fields.Further,the perturbations,u′and p′,areassumed to besmall and thenon-linear termsaredropped.The linearized perturbation equationsaregiven as:

    Here,σ′is the stress tensor for the perturbed solution.Eq.(4)subjected to the initial condition,u′(x,0)=u′0describes the evolution of small disturbances in the domain,?.Theboundary conditionson u′arehomogeneousversionsof thoseused for calculating thebaseflow(Eq.(2)).

    2.2 Global Linear Stability Analysis

    To conduct a global Linear stability analysis we assume the following form of the disturbancefield,u′and p′

    Substituting Eqs.(5)in the linearized disturbanceequations(Eqs.(4))we obtain:

    Eqs.(6)representsa generalized eigenvalue problem withλas the eigenvalue and(?u,?p)as the corresponding eigenmode.The boundary conditions for(?u,?p)are homogeneous version of those used for calculating the base flow(U,P).In general,the eigenvalue λ = λr+iλiis complex.The growth rate is given by the real part,λrof the eigenvalue whereas the imaginary part,λiis related to the temporal frequency of the of the disturbance field.A positive value ofλrindicates an unstable mode.This method has been utilized by several researchers in the past to investigatetheglobal linear stability of varioussteady flow configurations[Jackson(1987);Morzynski and Thiele(1991);Morzynski,Afanasiev,and Thiele(1999);Swaminathan,Sahu,Sameen,and Govindrajan(2011)].Mittal and Kumar(2003)proposed a stabilized fi nite element formulation for solving these equations and employed it to study theglobal stability propertiesof theflow past astationary and rotating cylinder.

    2.3 Local Stability Analysis:Orr-Sommerfeld Equation

    The disturbance field is assumed to be periodic along the two homogeneous directions:x and z.The wavenumbers along the x and z directions areαandβ,respectively.Thus,the perturbation fi eld in thisscenario isgiven by:

    Similar expressions can bewritten forwhich represent the x and z component of the disturbance fi eld.Let,k=α?i+β?k represent the wavenumber vector in the x?z planewith itsmagnitudegiven by k=.Substituting,Eq.(8)in thelinearized disturbance equation described by Eq.(7),weobtain:

    We consider the case when the streamwise wavenumber,α,is real and the eigenvalueλ =λr+iλiiscomplex.Thereal part,λr,isthegrowthrateof thedisturbance whileλi,theimaginary part,isthetemporal frequency of the disturbance.The disturbance associated with the eigenvalue that has the largest real mode is of major interest as it represents the fastest growing mode.For 2?D disturbances we can rewrite Eq.(9)to obtain the Orr-Sommerfeld(OS)equation:

    The disturbance velocity,u′,v′must vanish on the far-fi eld and solid boundaries,Γ.For the periodic disturbance fi eld considered this requires?u,?v to vanish onΓ.Using the continuity equation,one can simplify thisto:

    3 Formulation

    3.1 The Stabilized Finite Element Formulation for Global Linear Stability Analysis

    Let??R2be the spatial domain for global linear stability analysis(Eq.(6)).Consider afi niteelement discretization of?into subdomains?e,e=1,2,3,...,nel,where nelis the number of elements.Based on this discretization we define fi nite element trial function spaces for velocity and pressure perturbation fi elds asand,respectively.The weighting function space areand,respectively.Thesefunction spacesareselected by taking thehomogeneous Dirichlet boundary conditions into account,assubsetsof[H1h(?)]2and H1h(?),where H1h(?)isthe finitedimensional function spaceover?.Thestabilized finiteelement formulation of Eq.(6),is as follows:Findu?h∈Suuuhandp?h∈such that?w?h∈Vuuuhand

    Here,Uhrepresents the base flow at the element nodes.In the variational formulation given by Eq.(13),the first three terms constitute the Galerkin formulation of the problem.The terms involving the element level integrals are the stabilization terms added to the basic Galerkin formulation to enhance its numerical stability.These terms stabilize the computations against node-to-node oscillations in advection dominated flows and allow the use of equal-in-order basis functions for velocity and pressure.The terms with coeffi cientsτSUPGand τPSPGare based on the SUPG(Streamline-Upwind/Petrov-Galerkin)[Brooks and Hughes(1982)]and PSPG(Pressure-stabilized/Petrov-Galerkin)[Tezduyar,Mittal,Ray,and Shih(1992)]stabilizations.The SUPGformulation for convection dominated flowswas introduced by Hughes and Brooks(1979)and Brooks and Hughes(1982).PSPG stabilization for enabling the use of equal-order interpolations for the velocity and pressureto fluid flowsat finite Reynoldsnumber wasintroduced by Tezduyar,Mittal,Ray,and Shih(1992).The term with coeffi cientτLSICis a stabilization term based on theleast squares of thedivergencefreecondition on the velocity field.It providesnumerical stability at high Reynoldsnumber.Here,thestabilization coefficients used in the finite element formulation of LSA(Eq.(13))are computed on the basis of the base flow at the element nodes,Uh.The stabilization parameters aredefi ned as[Tezduyar,Mittal,Ray,and Shih(1992)]:

    Here,heis the element length based on the minimum edge length of an element[Mittal(2000)]and Uhisthebase flow velocity at element nodes.

    Eq.(13)lead to a generalized non-symmetric eigenvalue problem of the form A X?λB X=0.For our case,theeigenvalueproblem isslightly morecomplicated asthecontinuity equation responsiblefor determining pressurecausesthematrix B to becomesingular.Hence,to avoid singularity,wesolvetheinverseproblem,i.e.,eigenvalues for B X?μA X=0 are computed.Here,λ =1/μ.To check the stability of the steady-state solution we look for the rightmost eigenvalue(eigenvalue with largest real part),using thesubspaceiteration method[Stewart(1975)].

    3.2 The Spectral Method for Local Linear Stability Analysis

    Thespectral collocation method based on Chebyshev polynomialsof thefi rstkind[Schmid and Henningson(2001)]isused to solvethe Eq.(11)for carrying out thelocal sta-

    bility analysis.The Chebyshev polynomial of the fi rst kind isdefi ned as:

    for all non-negativeintegers n∈[0,N]and y∈[?1,1].By using asuitabletransformation,it ispossibleto map any other rangeof y to the Chebyshev domain[?1,1].The Chebyshev polynomials areutilized as the basis functions to approximate the eigenfunction,?v(y)in Eq.(8):

    Thisapproximation of theeigenfunction issubstituted in the OSequation(Eq.(11).It resultsin the following equation:

    Thecollocation method isemployed to evaluatetheconstants anin theapproximation given by Eq.(17).The following Gauss-Lobatto collocation pointsareused:

    Eq.(18)leads to the generalized eigenvalue problem of the form A X?λB X=0.In the present work,the numerical solution to the same is obtained using LAPACK[Anderson,Bai,Bischof,Blackford,Demmel,Dongarra,Du Croz,Greenbaum,Hammarling,McKenney,and Sorensen(1999)]libraries.

    4 Problem Setup

    4.1 The Base Flow

    Thelocal and theglobal linear stability analysisarecarried outfor theplane Poiseuille flow.Figure(1)shows the schematic of the flow.The fluid occupies the channel formed by two stationary plates parallel to each other and separated by a distance 2H.Theplatesarealigned with the x?axis.Thevelocity profilefor thebaseflow

    Figure1:Schematic of theplane Poiseuilleflow.

    is shown in the fi gure.It is parabolic and symmetric about the channel centerline.The equation for the streamwise component of velocity isgiven as:

    Here,H denotes half the channel width and Ucis the centerline velocity.All the lengthsarenon-dimensionalized with H,and velocity with Uc.The Reynoldsnumber,Re,isdefined as:

    where,νdenotesthekinematic viscosity of thefluid.

    4.2 Local Linear Stability Analysis

    The local analysis of the plane Poiseuille flow iscarried out via the solution to OS(Eq.(11)).The domain across the channel width,[?H,H],is mapped to[?1,1].No-slip boundary conditions are applied to the disturbance fi eld at the channel walls.In thissituation,Eq.(12)can berewritten as:

    The OSequation(Eq.(11)),along withtheboundary conditions(Eq.(22),issolved in thetemporal point of view.The wavenumber,α,is assumed to bereal.The OS equation is solved for different values of values ofαand Re.The effect of the number of grid points,along y,on the accuracy of the solution is investigated.It is found that 200 collocation points provide adequate spatial resolution.All the resultspresented in thispaper for the OSanalysisarewith 200 points.

    4.3 Global Linear Stability Analysis

    The flow in a fi nite streamwise length of the channel(=L)is considered for carrying out theglobal analysis.Thebaseflow isthefully developed steady flow in the channel.The streamwise velocity for the same is given by Eq.(20).The boundary conditions for thedisturbance fi eld are as follows.The disturbance velocity is prescribed a zero value at the upper and lower walls.To enable comparison with the local analysis,the disturbance is assumed to be periodic in the streamwise direction.Therefore,periodic boundary conditionsareapplied on all thevariablesat the inflow and theoutflow boundaries.Thefi niteelementmesh consistsof 24 elements alongthestreamwiseand 150elementsinthecross-flow directions.Thegrid points are uniformly spaced along x but are clustered close to the wall in the y direction.A mesh convergence study is carried out for the Re=7000 plane Poiseuille flow and L/2H=1.A more refi ned grid with roughly twice the resolution in each direction leadsto lessthan onepercentdifferencein theresults,thereby reflecting the adequacy of theoriginal fi nite element mesh.

    5 Results:Linear Stability Analysisof the Plane Poiseuille Flow

    5.1 OSAnalysis

    Local analysis via solution to the OS equation(Eq.(11))is carried out for various values of Re andα.At each(Re,α)the eigenvalue with the largest real part is identified.Figure(2)shows the variation of the growth rate of the disturbance associated with the rightmost eigenvalue with Re andα.The fi gure shows the iso-contours for various values of growth rate in the Re?αplane.The contour corresponding to zero growth rateistheneutral curve.Thecritical Re for theonset of instability is the lowest value of the Re on the neutral curve,for any value of α.The critical Re for this flow is found to be 5773,approximately and is marked in Figure(2).The value is in excellent agreement with results from earlier studies[Schmid and Henningson(2001)].

    Theresultsfor theflow at Re=7000 arepresented inmoredetail in Figure(3).This fi gureshowsthevariation of thereal(λr)and imaginary(λi)partsof therightmost eigenvalue with wavenumber(α)at Re=7000.While λrdenotes the growth rate,λiis related to the temporal frequency of the disturbance.We observe that the Re=7000 flow is linearly unstable only to disturbances whose wavenumber lies in a specifi c interval.The maximum growth rate is0.0017,approximately forα=1.00.

    Figure 2:Orr-Sommerfeld analysis of the Plane Poiseuille flow:iso-contours of constant growth rate.The critical Re for the onset of the instability of the flow is Recr=5773 and ismarked with a vertical broken line.

    Figure 3:Orr-Sommerfeld Analysis of the Plane Poiseuille Flow at Re=7000:variation of real and imaginary partof theright-most eigenvaluewith wavenumber,α.

    Figure 4:Global linear stability analysis of the Plane Poiseuille flow for Re=7000 and L/2H=5.10:the v′field for the eigenmodes corresponding to the two rightmost eigenvalues.The upper row corresponds to one cell in the domain(n=1)and has a growth rate,λr=?0.017.The lower row is for n=2 with two cells in thedomain;the growth ratefor this mode isλr=?0.0097.

    5.2 Global Analysis

    In thelocal analysis,the OSequation(Eq.(11))can besolved by usingαasoneof the independent variables.However,the global analysis(Eq.(6))does not directly offerαas an independent variable.The analysis,of course,can be carried out for different streamwise extent(L)of the computational domain.We attempt to understand the relation between L(for the global analysis)andα(for the local analysis).We propose that for a spatially periodic disturbance,its wavenumber is related to thelength of thecomputational domain as:

    where,n is the number of waves along the stream wise direction in the domain.To demonstratethis,weconsider theglobal linear stability analysisfor Re=7000.Fig.(4)shows the eigen modes associated with the two right most eigenvalues for L/2H=5.1.While the first one is associated with one wave(n=1),the other houses two waves(n=2)in the computational domain.Thus,they both represent different wavenumbersand areassociated with their own growth rates,aslisted in the caption of the fi gure.The real and imaginary part of the eigenvalue obtained from the global analysis,and their comparison with the values obtained from the local analysis,arealso shown in Figures(5)and(6).Thedatapointscorresponding to the two eigenmodes lie on the vertical line segment marked in the two figures for L/2H=5.10.The values from the local and global analysis are in excellent agreement.

    Figures(5)and(6)show the variation of the growth rate and the imaginary part of the rightmost eigenvalue from the global analysis for plane Poiseuille flow at Re=7000.The data points from the global analysis are marked by solid circles.Also shown in thesamefigurearetheresultsfrom thelocal analysis.Thevariation is associated with a number of peaks and valleys.We attempt to understand this behavior.It isdemonstrated in Fig.(4)that thecomputational domain may accommodate multiple cells of the disturbance.We fi rst identify in Figs.(5)and(6)the cases that are associated with onecell only(n=1)in thestreamwise extent of the domain.A best fi t to these points is in excellent agreement with the results from the local analysis.These curves are marked as L=2π/α in the figures.These curvescan also beutilized to understand thevariation ofλrandλiwithα.Wenote that thegrowth rateand temporal frequency of an eigenmodeshould depend onα,but must beindependent of thenumber of cellsof the sameαin the computational domain.Usingthisidea,and thedataforλrandλiv/sα fromthelocal analysis,the variation ofλrand λiwith L/2H is generated for multiple cells by observing that L=2πn/α,where n is the number of cells.These curves are shown in Figs.(5)and(6)for various values of n.The outer envelope of these curves is shown in thicker solid line.These curves provide an estimate of the variation of the rightmost eigenvaluewiththelength of thecomputational domain.Excellentagreement is observed between the estimated rightmost eigenvalue and the actual value from global LSA computationsfor n≥2.Wenotethatasthelengthof thecomputational domain isincreased,thedependenceof the growth rateof themost unstableeigenmodeon L becomesweaker.In theasymptotic limit of thedomain being infinitely long,the fastest growing mode comprises of infi nite cells of the n=1 eigenmode whose wavenumber is associated with largestλr.We also note from Fig.(5)that in certain situations it might be diffi cult to track the eigenmodes corresponding to low values ofαfrom the global analysis.Low values ofα correspond to large L/2H.Asseen from Fig.(5),at large L/2H,n=1 modeisnot necessarily theone with rightmost eigenvalue.For example,at L/2H=15 the rightmost eigenvalue corresponds to the mode with five cells(n=5).The modes with four,three,two and onecell have lower growth rate,and in the sameorder.Therefore,tracking the modefor n=1 for thisvalueof L/2H is relatively morechallenging than theones for higher valuesof n.

    To further demonstrate that the growth rate and temporal frequency of an eigenmode must be independent of the number of streamwise cells in the global analysis,weconsider thecasewhereweseek therightmost eigenvalueforα=1.05.For n=1,thiscorrespondsto L/2H=3.0,approximately.Figure(7)showstheeigenmodesfromtheglobal analysisfor variousvaluesof L/2H for thesameα(=1.05).Thevariousvaluesof L arechosen by varying n in therelation L=2 nπ/α.A broken horizontal lineismarked in Figures(5)and(6)to show thereal and imaginary partof therightmosteigenvaluefor variousvaluesof L thatcorrespond toα=1.05.We observe that all these modes are associated with the same eigenvalue.In fact,theeigenmodesarealso of thesamefamily.They areshown in Figure(7)and have

    Figure 5:Variation of the growth rate of the leading eigenvalue with L/2H for the plane Poiseuilleflow for Re=7000:thesolid dotsrepresent thegrowth rateof the mostunstablemodeobtained atvariousvaluesof L/2H fromglobal LSA.Thesolid(red)curveisobtained from thelocal(Orr-Sommerfeld)analysis.It isin excellent agreement with the best fi t to the points corresponding to one streamwise wave(n=1)from global analysis asper the relation L=2π/α.The curve isreplicated for various n to show the predicted variation ofλr with L,for the global analysis using the relation L=n(2π/α),when the domain houses different number of cells.Theouter envelopeof thesecurves,showninthicker solid line,representsthe eigenmode associated with the rightmost eigenvalue for the corresponding length of thecomputational domain.

    the sameflow structure,albeit with different number of cells.

    6 Concluding Remarks

    Hydrodynamic stability of shear flows has been widely investigated in the past usinglocal and global Linear Stability Analysis(LSA).Inthiswork wehavereviewed thetwo approachesand attempted to highlightthedifferencebetween thetwo in the context of their application to parallel shear flows.Resultsfor thelinear stability of plane Poiseuille flow have been presented,using both approaches.The local analysisiscarried out by solving the Orr-Sommerfeld(OS)equation using thespectral collocation method based on Chebyshev polynomials.The analysis has been carried out for various wavenumbers,αof the streamwise periodic disturbance fi eld.The critical Re for the onset of linear instability for plane Poiseuille flow is found to be 5773,which is in good agreement with earlier results[Schmid and Henningson(2001)].The stability of the flow at Re=7000 has been presented in more detail.For example,the variation of the real and imaginary part of the least stable eigenvalue withαhas been presented.Unlike the local analysis which involves solution to an ordinary differential equation,the global analysis involves fi nding solution to a set of partial differential equations.The analysis has been carried out for atwo-dimensional disturbance fi eld that isassumed to bespatially periodic along the stream wise direction.A stabilized finite element method has been presented for carrying out the global LSA in primitive variables.Equal-in-order fi nite element functions are used for representing velocity and pressure.To suppress the numerical oscillationsthat might appear in thecomputations,the SUPGand PSPG,stabilizationsareadded tothe Galerkinfiniteelementformulation.Theformulation hasbeen used to carry out the linear stability analysisfor the plane Poiseuille flow at Re=7000.Computations are carried out for various values of the streamwise length,L,of thecomputational domain.

    Figure 6:Variation of the imaginary part of theleading eigenvalue with L/2H for theplane Poiseuilleflow for Re=7000:thesolid dotsrepresent theimaginary part of the most unstable mode obtained at various values of L/2H from global LSA.The solid(red)curve isobtained from the local(Orr-Sommerfeld)analysis.It isin excellent agreement with thebest fit to thepointscorresponding to onestreamwise wave(n=1)from global analysis as per the relation L=2π/α.The curve is replicated for various n to show the predicted variation ofλi with L,for the global analysisusingtherelation L=n(2π/α),when thedomainhousesdifferent number of cells.The curves shown in thicker solid line representsλi associated with the rightmost eigenvaluefor thecorresponding length of thecomputational domain.

    Figure 7:Eigenmodes of v′corresponding to the leading eigenvalue for various lengths of the domain obtained with the global LSA for the plane Poiseuille flow for Re=7000 for disturbancesthat areperiodic in thestreamwise direction.

    Unlike the local analysis, the global analysis can handle non-periodic disturbances and is applicable to non-parallel flows as well. However, the global analysis is signifi cantly more expensive than the local a nalysis. For the parallel flow and with spatially periodic disturbances the present work brings out a very interesting relationship between the wave number of the disturbance and the streamwise extent of the domain in the global analysis. When the eigenmode contains only once cell, the results from the local and global analysis are virtually identical; the wavenumber and streamwise extent of the domain are related as α = 2 π/L. However, when the eigenmode consists of n cells along the streamwise length of the domain the relationship is: α = (2 πn)/L. For a very large value of L, the global analysis results in an eigenmode with a large number of cells of the eigenmode whose α corresponds to the mode with largest growth rate. If one would like to use the global analysis to create the growth rate v/s α curve for the rightmost eigenvalue, as is done in the local analysis for a specific value of Re, the procedure is complicated by the number of cells that are housed in the domain. In the scenario when L is relatively large, to track an eigenmode for low α, the eigenmode associated with one cell might not be the most unstable mode. Therefore, one needs to examine the eigenmodes for the first few eigenvalues that are arranged in the descending order of their real part.The one that corresponds to α = 2 π/L is the eigenmode which consists of only one cell along the streamwise direction.

    Acknowledgement:The help from Mr.Hardik Parwana in carrying out some of thecomputationsisgratefully acknowledged.

    Anderson,E.;Bai,Z.;Bischof,C.;Blackford,S.;Demmel,J.;Dongarra,J.;Du Croz,J.;Greenbaum,A.;Hammarling,S.;McKenney,A.;Sorensen,D.(1999):LAPACKUsers’Guide.Society for Industrial and Applied Mathematics,Philadelphia,PA,third edition.

    Boiko,A.V.;Dovgal,A.V.;Grek,G.R.;Kozlov,V.V.(2012): Physics of Transitional Shear Flows.Springer-Verlag.

    Brooks,A.;Hughes,T.(1982):Streamlineupwind/petrov-galerkin formulations for convection dominated flows with particular emphasis on the incompressible navier-stokes equations.Computer Methods in Applied Mechanics and Engineering,vol.32,pp.199–259.

    Chandrasekhar,S.(1981): Hydrodynamic and hydromagnetic stability.Dover.

    Chomaz,J.-M.(2005): Global instabilities in spatially developing flows:nonnormality and nonlinearity.Annual Review of Fluid Mech.,vol.37,pp.357–392.

    Chomaz,J.M.;Huerre,P.;Redekopp,L.G.(1988):Bifurcations to local and global modes in spatially developing flows.Physical Review Letters,vol.60,pp.25–28.

    Davey,A.;Drazin,P.(1969):Thestability of poiseuilleflow in apipe.J.Fluid Mech.,vol.36,pp.209–218.

    Garg, V. K.; Rouleau, W. T. (1972): Linear spatial stability of pipe poiseuille flow. J. Fluid Mech., vol. 54, pp. 113–127.

    Gaster,M.(1962): A note on the relation between temporally-increasing and spatially-increasing disturbances in hydrodynamic stability.J.Fluid Mech.,vol.14,pp.222–224.

    Huerre,P.(2000): Open shear flow instabilities.In Batchelor,G.;Moffatt,H.;Worster,M.(Eds):Perspectivesin Fluid Dynamics,pp.159–229.Cambridge.

    Huerre,P.;Monkewitz,P.(1990): Local and global instabilities in spatially developing flows.Annual Review of Fluid Mech.,vol.22,pp.473–537.

    Hughes,T.;Brooks,A.(1979): A multi-dimensional upwind scheme with no crosswind diffusion.Journal of Applied Mechanics,vol.34,pp.19–35.

    Jackson,C.(1987):A fi niteelement study of theonset of vortex shedding in flow past variously shaped bodies.J.Fluid Mech.,vol.182,pp.23.

    Mittal,S.(2000): On the performance of high aspect-ratio elements for incompressible flows.Computer Methods in Applied Mechanics and Engineering,vol.188,pp.269–287.

    Mittal,S.(2004):Three-dimensional instabilitiesin flow past a rotating cylinder.Journal of Applied Mechanics,vol.71,pp.89–95.

    Mittal,S.;Kumar,B.(2003): Flow past a rotating cylinder.Journal of Fluid Mechanics,vol.476,pp.303–334.

    Monkewitz,P.A.(1988): The absolute and convective nature of instability in two-dimensional wakes at low Reynolds numbers.Physics of Fluids,vol.31,pp.999–1006.

    Morzynski, M.; Afanasiev, K.; Thiele, F. (1999): Solution of the eigenvalue problems resulting from global non-parallel flow s ta bility analysis.Comput. Meth-ods Appl. Mech. Eng., vol. 169, pp. 161.

    Morzynski,M.;Thiele,F.(1991):Numerical stability analysis of aflow about a cylinder.Z.Angew.Math.Mech.,vol.71,pp.T424.

    Navrose;Meena,J.;Mittal,S.(2015): Three-dimensional flow past a rotating cylinder.J.Fluid Mech.,vol.766,pp.28–53.

    Orr,W.M.(1907):The stability or instability of the steady motions of a perfect liquid and of a viscousliquid.Proc.R.Irish Acad.Sec.A,vol.27,pp.9–138.

    Orszag,S.A.(1971):Accurate solution of the orr-sommerfeld stability equation.J.Fluid Mech.,vol.50,pp.689–703.

    Pierrehumbert,R.T.(1985): Local and global baroclinic instability of zonally varying flow.Journal of the Atmospheric Sciences,vol.41,pp.2141–2162.

    Saraph,V.;Vasudeva,B.R.;Panikar,J.(1979):Stability of parallel flowsby the fi nite element method.Int.J.Numer.Methods Engineering,vol.17,pp.853–870.

    Schmid,P.J.;Henningson,D.S.(2001): Stability and Transition in Shear Flows.Springer-Verlag.

    Sommerfeld,A.(1908):Ein Beitrag zur hydrodynamischen Erkl?erung der turbulenten Flüessigkeitsbewegungen. Proc.Fourth Internat.Cong.Math.,Rome,vol.III,pp.116–128.

    Stewart,G.(1975):Methods of simultaneous iteration for calculating eigenvectors of matrices.In Miller,J.(Ed):Topics in Numerical Analysis II,pp.169–185.Academic Press:New York.

    Swaminathan,G.;Sahu,K.;Sameen,A.;Govindrajan,R.(2011): Global instabilities in diverging channel flows.Theor.Comput.Fluid Dyn.,vol.25,pp.53–64.

    Tezduyar,T.;Mittal,S.;Ray,S.;Shih,R.(1992):Incompressibleflow computations with stabilized bilinear and linear equal-order-interpolation velocity-pressure elements.Comput.Meth.Appl.Mech.Engrg,vol.95,pp.221.

    Theofilis,V.(2011):Global linear instability.Annual Review of Fluid Mech.,vol.43,pp.319–352.

    Verma,A.;Mittal,S.(2011): A new unstable mode in the wake of a circular cylinder.Phys.Fluids.,vol.23,pp.121701.

    Yang,X.;Zebib,A.(1989): Absolute and convective instability of a cylinder wake.Physicsof Fluids A,vol.1,pp.689–696.

    脱女人内裤的视频| 欧美大码av| 欧美日韩国产亚洲二区| 日韩免费av在线播放| 免费av毛片视频| 狠狠狠狠99中文字幕| 少妇人妻一区二区三区视频| 俄罗斯特黄特色一大片| 久久久久性生活片| 国产熟女xx| 亚洲专区中文字幕在线| 国产三级黄色录像| 久久久国产成人免费| 国产精品永久免费网站| 成人高潮视频无遮挡免费网站| 99国产极品粉嫩在线观看| 床上黄色一级片| 香蕉国产在线看| 91大片在线观看| 亚洲欧美日韩东京热| 在线观看www视频免费| 久久久久国产精品人妻aⅴ院| 91国产中文字幕| av超薄肉色丝袜交足视频| 嫁个100分男人电影在线观看| 男插女下体视频免费在线播放| 日本一区二区免费在线视频| 精品欧美一区二区三区在线| 级片在线观看| 一级毛片高清免费大全| 欧美激情久久久久久爽电影| 日本黄色视频三级网站网址| 国产精品久久久久久亚洲av鲁大| 男女午夜视频在线观看| 国产蜜桃级精品一区二区三区| 国产黄a三级三级三级人| 国产高清有码在线观看视频 | 脱女人内裤的视频| 欧美又色又爽又黄视频| 国内毛片毛片毛片毛片毛片| 久久草成人影院| 国产精品亚洲美女久久久| 久久久久亚洲av毛片大全| av有码第一页| 国产69精品久久久久777片 | 男女下面进入的视频免费午夜| 两个人视频免费观看高清| 又爽又黄无遮挡网站| 9191精品国产免费久久| 丁香欧美五月| 国产成人精品久久二区二区91| 成年女人毛片免费观看观看9| 搡老熟女国产l中国老女人| 久久久久亚洲av毛片大全| 亚洲人成77777在线视频| 国产麻豆成人av免费视频| 日日摸夜夜添夜夜添小说| 每晚都被弄得嗷嗷叫到高潮| 一个人免费在线观看的高清视频| 久久久精品国产亚洲av高清涩受| 欧美绝顶高潮抽搐喷水| 日本 av在线| 亚洲aⅴ乱码一区二区在线播放 | 女生性感内裤真人,穿戴方法视频| 99在线视频只有这里精品首页| 久久久精品大字幕| 好看av亚洲va欧美ⅴa在| 50天的宝宝边吃奶边哭怎么回事| 国产亚洲欧美在线一区二区| 变态另类丝袜制服| 亚洲七黄色美女视频| 欧美乱妇无乱码| 成年女人毛片免费观看观看9| 欧美一级a爱片免费观看看 | 国产精品,欧美在线| 国产精品自产拍在线观看55亚洲| 日本 欧美在线| 老司机午夜十八禁免费视频| 中文在线观看免费www的网站 | 露出奶头的视频| 久久久久国内视频| 脱女人内裤的视频| 天天躁狠狠躁夜夜躁狠狠躁| 色精品久久人妻99蜜桃| 丰满人妻熟妇乱又伦精品不卡| 一二三四在线观看免费中文在| 99久久精品热视频| 别揉我奶头~嗯~啊~动态视频| 国产精品野战在线观看| 丁香欧美五月| 国产亚洲av嫩草精品影院| 日韩大尺度精品在线看网址| 欧美又色又爽又黄视频| 日韩欧美免费精品| 制服人妻中文乱码| 欧美+亚洲+日韩+国产| a级毛片a级免费在线| 欧美色欧美亚洲另类二区| 特级一级黄色大片| 免费电影在线观看免费观看| 欧美人与性动交α欧美精品济南到| 成人一区二区视频在线观看| 无遮挡黄片免费观看| 狠狠狠狠99中文字幕| 日本在线视频免费播放| 国产一级毛片七仙女欲春2| 长腿黑丝高跟| 成年免费大片在线观看| 天天一区二区日本电影三级| 男女做爰动态图高潮gif福利片| 免费看十八禁软件| 男男h啪啪无遮挡| 99国产极品粉嫩在线观看| 精品免费久久久久久久清纯| 亚洲男人的天堂狠狠| 麻豆成人av在线观看| 免费在线观看完整版高清| 国产av又大| 麻豆一二三区av精品| 亚洲真实伦在线观看| 国产黄色小视频在线观看| 亚洲精品国产一区二区精华液| 熟女电影av网| 精品久久久久久久毛片微露脸| 夜夜夜夜夜久久久久| 亚洲一区中文字幕在线| 国产精品一区二区三区四区久久| 最新在线观看一区二区三区| 亚洲精品一卡2卡三卡4卡5卡| 97碰自拍视频| cao死你这个sao货| 国产探花在线观看一区二区| 国产男靠女视频免费网站| 久久精品国产亚洲av高清一级| 蜜桃久久精品国产亚洲av| 国产亚洲av高清不卡| 天堂动漫精品| 免费看日本二区| 婷婷精品国产亚洲av| 波多野结衣高清作品| 三级毛片av免费| 亚洲专区字幕在线| 免费在线观看完整版高清| 99在线视频只有这里精品首页| 国产精品电影一区二区三区| 妹子高潮喷水视频| 国产亚洲精品第一综合不卡| 欧美大码av| 国产99久久九九免费精品| 又黄又爽又免费观看的视频| 成人国产一区最新在线观看| 波多野结衣高清作品| 久久精品成人免费网站| √禁漫天堂资源中文www| 两性午夜刺激爽爽歪歪视频在线观看 | 免费无遮挡裸体视频| 亚洲乱码一区二区免费版| 久久精品综合一区二区三区| av福利片在线| 黑人欧美特级aaaaaa片| 一级黄色大片毛片| 国产蜜桃级精品一区二区三区| 亚洲国产精品成人综合色| 国产aⅴ精品一区二区三区波| 12—13女人毛片做爰片一| 欧美日韩亚洲综合一区二区三区_| 成人亚洲精品av一区二区| 中文亚洲av片在线观看爽| 床上黄色一级片| 可以免费在线观看a视频的电影网站| 亚洲欧美精品综合一区二区三区| 亚洲人成77777在线视频| 欧美中文综合在线视频| 久久久久久免费高清国产稀缺| 久久精品综合一区二区三区| 首页视频小说图片口味搜索| 三级国产精品欧美在线观看 | 免费高清视频大片| 国产成人欧美在线观看| 精品不卡国产一区二区三区| av有码第一页| 成人国语在线视频| 日本熟妇午夜| 久久精品综合一区二区三区| 欧美人与性动交α欧美精品济南到| 国产成人av激情在线播放| av天堂在线播放| 一级片免费观看大全| 人妻丰满熟妇av一区二区三区| 免费看a级黄色片| 免费看十八禁软件| 99国产精品一区二区三区| 久久精品国产清高在天天线| 欧美中文日本在线观看视频| 12—13女人毛片做爰片一| 亚洲欧美一区二区三区黑人| 国产成人av激情在线播放| 国内少妇人妻偷人精品xxx网站 | e午夜精品久久久久久久| 99精品久久久久人妻精品| 国产亚洲欧美98| 天堂av国产一区二区熟女人妻 | 美女高潮喷水抽搐中文字幕| 男人的好看免费观看在线视频 | 91麻豆精品激情在线观看国产| 中国美女看黄片| svipshipincom国产片| 国产精品一区二区三区四区免费观看 | 男女下面进入的视频免费午夜| 亚洲av第一区精品v没综合| 中文在线观看免费www的网站 | 国产成人啪精品午夜网站| 老熟妇仑乱视频hdxx| 亚洲一码二码三码区别大吗| 看黄色毛片网站| 欧美黑人精品巨大| 午夜激情福利司机影院| 欧美精品亚洲一区二区| or卡值多少钱| 高潮久久久久久久久久久不卡| 亚洲成人精品中文字幕电影| 亚洲成人久久爱视频| 欧美精品亚洲一区二区| 午夜亚洲福利在线播放| 色综合婷婷激情| 国产精品一及| 精品久久久久久成人av| 国产野战对白在线观看| 成年人黄色毛片网站| 免费在线观看视频国产中文字幕亚洲| 亚洲精品粉嫩美女一区| 国产aⅴ精品一区二区三区波| 最好的美女福利视频网| 成人av在线播放网站| 国产精品一区二区三区四区免费观看 | 老熟妇仑乱视频hdxx| 国产成人精品久久二区二区91| 免费观看精品视频网站| 日韩欧美一区二区三区在线观看| 久久久国产成人精品二区| 国产午夜福利久久久久久| 日日干狠狠操夜夜爽| 国产av麻豆久久久久久久| 男女下面进入的视频免费午夜| 一进一出抽搐动态| 不卡一级毛片| 免费看十八禁软件| 白带黄色成豆腐渣| 一个人免费在线观看电影 | ponron亚洲| av超薄肉色丝袜交足视频| netflix在线观看网站| 亚洲成人国产一区在线观看| 欧洲精品卡2卡3卡4卡5卡区| 亚洲国产精品999在线| 真人一进一出gif抽搐免费| 久久久久国产精品人妻aⅴ院| 国产精品九九99| 精品国产乱子伦一区二区三区| 国产野战对白在线观看| 午夜激情福利司机影院| 午夜视频精品福利| 日本一二三区视频观看| 国产精品98久久久久久宅男小说| 51午夜福利影视在线观看| 亚洲 欧美一区二区三区| 久久这里只有精品中国| 免费看日本二区| 天天躁夜夜躁狠狠躁躁| 日本免费a在线| 欧洲精品卡2卡3卡4卡5卡区| www.www免费av| 午夜精品久久久久久毛片777| 国内久久婷婷六月综合欲色啪| 岛国在线观看网站| 男女做爰动态图高潮gif福利片| 国产成人影院久久av| 观看免费一级毛片| 婷婷精品国产亚洲av在线| 麻豆av在线久日| 精品国产乱子伦一区二区三区| 国产三级中文精品| 日本免费a在线| svipshipincom国产片| 国产99白浆流出| 久久久久久国产a免费观看| 性欧美人与动物交配| 激情在线观看视频在线高清| 国产精品一及| 亚洲成人久久爱视频| 人人妻人人看人人澡| 一级a爱片免费观看的视频| 亚洲av五月六月丁香网| 曰老女人黄片| 国产野战对白在线观看| 在线观看美女被高潮喷水网站 | 亚洲男人天堂网一区| 老鸭窝网址在线观看| 欧美色欧美亚洲另类二区| 亚洲欧美日韩东京热| 久久久久久九九精品二区国产 | 国产又黄又爽又无遮挡在线| 两个人的视频大全免费| 90打野战视频偷拍视频| 中出人妻视频一区二区| 国产精品1区2区在线观看.| 免费高清视频大片| 欧美中文日本在线观看视频| 一边摸一边做爽爽视频免费| 午夜福利免费观看在线| 亚洲男人的天堂狠狠| 亚洲中文字幕一区二区三区有码在线看 | 免费av毛片视频| 成人三级做爰电影| 亚洲国产看品久久| 熟女少妇亚洲综合色aaa.| 免费看a级黄色片| 黄色片一级片一级黄色片| 麻豆久久精品国产亚洲av| 精品久久久久久久末码| 国产乱人伦免费视频| 免费看美女性在线毛片视频| 女人爽到高潮嗷嗷叫在线视频| 麻豆国产97在线/欧美 | 在线永久观看黄色视频| 蜜桃久久精品国产亚洲av| 巨乳人妻的诱惑在线观看| 99久久精品国产亚洲精品| 成人国产一区最新在线观看| 无人区码免费观看不卡| 国产精品免费视频内射| av有码第一页| 久久亚洲真实| 青草久久国产| 国产激情偷乱视频一区二区| 日本 av在线| 久9热在线精品视频| 日日爽夜夜爽网站| 波多野结衣高清无吗| 人成视频在线观看免费观看| 亚洲第一电影网av| 我的老师免费观看完整版| 久久久久久久午夜电影| 精品日产1卡2卡| 日本成人三级电影网站| videosex国产| 法律面前人人平等表现在哪些方面| 在线a可以看的网站| 久久久久久大精品| 一卡2卡三卡四卡精品乱码亚洲| 999久久久精品免费观看国产| 色综合站精品国产| 成人亚洲精品av一区二区| 免费在线观看影片大全网站| 国产高清有码在线观看视频 | 成年版毛片免费区| 久久久久久大精品| 成人18禁高潮啪啪吃奶动态图| 午夜免费成人在线视频| 黄片大片在线免费观看| 免费在线观看视频国产中文字幕亚洲| 麻豆国产97在线/欧美 | 两性午夜刺激爽爽歪歪视频在线观看 | 一个人免费在线观看的高清视频| 国产日本99.免费观看| 色老头精品视频在线观看| 岛国视频午夜一区免费看| 亚洲欧美激情综合另类| 变态另类成人亚洲欧美熟女| 亚洲熟妇熟女久久| 国产成人欧美在线观看| 亚洲精品在线美女| 免费看美女性在线毛片视频| 精品第一国产精品| 免费看a级黄色片| 男人的好看免费观看在线视频 | 色综合站精品国产| 一区二区三区国产精品乱码| 久久中文字幕人妻熟女| 欧美性长视频在线观看| 欧美黑人巨大hd| 免费无遮挡裸体视频| 老鸭窝网址在线观看| 亚洲欧美精品综合一区二区三区| 免费看十八禁软件| 不卡一级毛片| 热99re8久久精品国产| 人人妻,人人澡人人爽秒播| 99热6这里只有精品| 婷婷精品国产亚洲av在线| 亚洲成a人片在线一区二区| 99国产极品粉嫩在线观看| 国产区一区二久久| 国产真人三级小视频在线观看| 欧美极品一区二区三区四区| 国产蜜桃级精品一区二区三区| 日韩欧美国产在线观看| 欧美黄色片欧美黄色片| 日日摸夜夜添夜夜添小说| 欧美一区二区国产精品久久精品 | 波多野结衣高清作品| 啦啦啦观看免费观看视频高清| 亚洲精品粉嫩美女一区| 久久久久免费精品人妻一区二区| 黄频高清免费视频| 国产精品免费视频内射| 亚洲国产日韩欧美精品在线观看 | 亚洲成人国产一区在线观看| 免费在线观看日本一区| 琪琪午夜伦伦电影理论片6080| 脱女人内裤的视频| 舔av片在线| 国产亚洲精品一区二区www| 国产伦人伦偷精品视频| 国产激情偷乱视频一区二区| 免费电影在线观看免费观看| 亚洲自拍偷在线| 香蕉丝袜av| 最近最新中文字幕大全电影3| 99国产极品粉嫩在线观看| a在线观看视频网站| 国产成+人综合+亚洲专区| 精品国产美女av久久久久小说| 两人在一起打扑克的视频| 一边摸一边做爽爽视频免费| 搡老妇女老女人老熟妇| 91老司机精品| 亚洲国产精品sss在线观看| 麻豆av在线久日| 熟妇人妻久久中文字幕3abv| 99在线人妻在线中文字幕| 亚洲国产精品成人综合色| 中文字幕av在线有码专区| 天天添夜夜摸| 老熟妇乱子伦视频在线观看| 欧美乱妇无乱码| 男人舔女人的私密视频| 女人被狂操c到高潮| 久久人人精品亚洲av| 亚洲av电影不卡..在线观看| 亚洲成人国产一区在线观看| 精品久久久久久久人妻蜜臀av| 极品教师在线免费播放| 亚洲av五月六月丁香网| 久久午夜亚洲精品久久| av中文乱码字幕在线| 亚洲第一欧美日韩一区二区三区| 国产精品久久久人人做人人爽| 黄色a级毛片大全视频| 麻豆国产97在线/欧美 | 亚洲精品一区av在线观看| 精品午夜福利视频在线观看一区| 宅男免费午夜| 中文字幕高清在线视频| 成人午夜高清在线视频| 熟女电影av网| 久久香蕉激情| 国产成人系列免费观看| 最近最新中文字幕大全电影3| 亚洲精品粉嫩美女一区| 日韩国内少妇激情av| 中国美女看黄片| 少妇粗大呻吟视频| 中出人妻视频一区二区| 精品国产美女av久久久久小说| 国产探花在线观看一区二区| 少妇裸体淫交视频免费看高清 | 丝袜美腿诱惑在线| 亚洲专区国产一区二区| 国产熟女午夜一区二区三区| 无遮挡黄片免费观看| 丰满人妻熟妇乱又伦精品不卡| 深夜精品福利| 黑人操中国人逼视频| 免费人成视频x8x8入口观看| 国产午夜福利久久久久久| 99热6这里只有精品| 99re在线观看精品视频| 夜夜躁狠狠躁天天躁| 黄色视频,在线免费观看| 午夜激情av网站| 两性夫妻黄色片| 国产av在哪里看| 午夜福利成人在线免费观看| 深夜精品福利| 久久精品亚洲精品国产色婷小说| 国产成人影院久久av| 伊人久久大香线蕉亚洲五| 久久精品国产99精品国产亚洲性色| 久久婷婷人人爽人人干人人爱| 欧美最黄视频在线播放免费| 国产黄色小视频在线观看| 久久国产精品影院| 日本一区二区免费在线视频| 国产不卡一卡二| 高潮久久久久久久久久久不卡| 欧美性猛交╳xxx乱大交人| 成年版毛片免费区| 精品熟女少妇八av免费久了| 国产不卡一卡二| 中文字幕久久专区| а√天堂www在线а√下载| 日本黄色视频三级网站网址| 亚洲人成伊人成综合网2020| 久久这里只有精品中国| 看片在线看免费视频| 大型av网站在线播放| 身体一侧抽搐| 91麻豆av在线| 哪里可以看免费的av片| 两性午夜刺激爽爽歪歪视频在线观看 | 欧美乱码精品一区二区三区| 最近最新免费中文字幕在线| 男人舔女人的私密视频| √禁漫天堂资源中文www| 国产午夜精品久久久久久| 两性午夜刺激爽爽歪歪视频在线观看 | 两性午夜刺激爽爽歪歪视频在线观看 | 在线十欧美十亚洲十日本专区| 99riav亚洲国产免费| 好男人电影高清在线观看| 亚洲avbb在线观看| 国产三级黄色录像| 午夜激情av网站| bbb黄色大片| 国产又色又爽无遮挡免费看| 日日夜夜操网爽| 欧美成人性av电影在线观看| 每晚都被弄得嗷嗷叫到高潮| 亚洲av电影不卡..在线观看| 亚洲精品国产精品久久久不卡| 婷婷亚洲欧美| 亚洲人与动物交配视频| 99久久精品热视频| 中文字幕av在线有码专区| 一级片免费观看大全| 亚洲全国av大片| 亚洲国产精品久久男人天堂| 91成年电影在线观看| 亚洲精品一卡2卡三卡4卡5卡| www.www免费av| 在线观看免费视频日本深夜| 中亚洲国语对白在线视频| 成年版毛片免费区| 亚洲国产精品成人综合色| 麻豆成人av在线观看| 久久久久九九精品影院| 老司机靠b影院| 久久久国产成人免费| 成人av一区二区三区在线看| 狂野欧美白嫩少妇大欣赏| 午夜福利在线在线| 又爽又黄无遮挡网站| 午夜福利欧美成人| 国产久久久一区二区三区| 黄片小视频在线播放| 免费在线观看日本一区| 露出奶头的视频| 日韩国内少妇激情av| 一级片免费观看大全| 国产黄片美女视频| 1024视频免费在线观看| 欧美三级亚洲精品| 麻豆成人午夜福利视频| 国产三级中文精品| 性欧美人与动物交配| 亚洲片人在线观看| 最近最新中文字幕大全电影3| 天堂动漫精品| 国产1区2区3区精品| 国内毛片毛片毛片毛片毛片| 日韩国内少妇激情av| 久久天躁狠狠躁夜夜2o2o| 身体一侧抽搐| 精品一区二区三区四区五区乱码| 亚洲美女视频黄频| 亚洲成av人片在线播放无| 国产亚洲av嫩草精品影院| 啦啦啦免费观看视频1| 国产激情偷乱视频一区二区| 一边摸一边抽搐一进一小说| 亚洲欧美日韩无卡精品| 亚洲 欧美一区二区三区| 国产精品 国内视频| 神马国产精品三级电影在线观看 | 国产亚洲欧美在线一区二区| 亚洲午夜精品一区,二区,三区| 非洲黑人性xxxx精品又粗又长| 美女黄网站色视频| 欧美 亚洲 国产 日韩一| 制服诱惑二区| 欧美黑人精品巨大| 国产午夜精品论理片| 午夜精品久久久久久毛片777| 久久婷婷人人爽人人干人人爱| 香蕉久久夜色| 一级作爱视频免费观看| 国产精品久久久久久精品电影| 搡老熟女国产l中国老女人| 欧美色视频一区免费| 亚洲美女视频黄频| 欧美日韩国产亚洲二区| 99精品久久久久人妻精品| 国产欧美日韩一区二区三| 又黄又粗又硬又大视频| 麻豆成人午夜福利视频| 国产欧美日韩一区二区三| 国产精品乱码一区二三区的特点| 国产麻豆成人av免费视频| 国产精品香港三级国产av潘金莲| 真人做人爱边吃奶动态| 午夜福利视频1000在线观看| 在线观看一区二区三区| 久久久久免费精品人妻一区二区| 国产熟女午夜一区二区三区|