• <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 ?

    Shear law of velocity boundary layer

    2016-12-12 02:50:52,
    關(guān)鍵詞:邊界層商事定律

    ,

    (College of Mathematics and Systems Science, Shenyang Normal University, Shenyang 110034, China)

    ?

    Shear law of velocity boundary layer

    LIMingjun,CAIZhenyu

    (College of Mathematics and Systems Science, Shenyang Normal University, Shenyang 110034, China)

    The theory of boundary layer has been known for many years since 1900’s due to Ludwig Prandtl’s contribution. However, solutions to the boundary layer equation does not properly describe high-Reynolds-number flows. In this paper, a generalized Blasius’s equation with fluctuation functionFis derived by using the Prindtl’s boundary equation of mean value and fluctuation, and a shear law of velocity in the inner boundary layer is built theoretically and numerically. The boundary thickness of the leading edge point is found to beδ0=c2/Reu, whereReu=U∞/ν,c=1.7208 as displacement constant orc=0.664 as momentum constant, respectively. Moreover, the Limit Value Theorem on the velocity boundary layer and numerical experiments show that the Newtonian linear shear law of velocity boundary layer is perfectly satisfied forF=0.1 andF=0.01, and the nonlinear shear law is presented forF=0.001 andF→0. Such a mechanism has never been demonstrated in the classical boundary layer theory.

    generalized Blasius’s equation; fluctuation function; boundary thickness; Newtonian linear shear law

    0 Introduction

    A central issue in the study of fluid dynamics is to determine the velocity profile and its valid thickness within a very thin layer in the neighborhood of the solid wall, i.e., the viscous boundary layer theory[1]. In 1904, the concept and basic equations of the laminar boundary layer theory were first formulated by Prandtl[1-2]. Then, this problem was solved numerically by Blasius[3]. If neglecting the stream direction pressure gradient of the Prandtl’s boundary layer equations, one can get the well-known Blasius's equation

    (1)

    The above equation is an autonomous, third order, nonlinear ordinary differential equation, which results from three similar boundary conditions as follows.

    (2)

    Conditionsf=0 andf′=0 on the solid wall are “no-slip” boundary conditions, which are fundamental elements in analyzing continuum fluid flow. The analytic evaluation of the Blasius’s equation is quite tedious, sometimes resorts to numerical tools. As Blasius has done, similarity transformation technique play a significant role in most studies in solving the boundary layer problem by transforming a partial differential equation (PDE) into an ordinary differential equation (ODE)[4-5]. However, the simulation of fluid flows with “no-slip” boundaries is plagued with difficulties of rough hydrophobic surfaces, so-called “superhydrophobic” surfaces, and molecular dynamics simulations with consideration of the “slip” conditions[6-8]are usually very time consuming. In 1968, Steinheuer[9]gave a systematic review of the solutions to Blasius's equation.

    In the boundary layer, the transition process from laminar to turbulent flow is a basic scientific problem in modern fluid mechanics, and different concepts have been developed to explain the transition mechanism[10]. Among these concepts, streaks and streamwise vorticity are two key features. Some recent works suggested the existence of a self-sustaining process between streamwise vortices and streaks at Reynolds numbers[10-13]. Duriez's study[12]shows that, for Reynolds numberReh=U∞h/v=900(h(h=6 mm)is the cylinder height), the instantaneous velocity field is very fluctuating, and a streamwise modulation exists. He suggested that the self sustaining mechanism through destabilization of the streamwise streaks is probably the reason for the fluctuating at the largest Reynolds numbers where the flow undergoes a transition toward a weakly turbulent state.

    As well known, the boundary layer flow takes place mainly parallel to the solid surface, and the inner boundary layer velocity varies rapidly normal to the surface and slowly along the stream direction. Thus the pressure in the boundary layer can be simply assumed as the pressurep(x). Despite all those features, there has not been a deliberate effort to design the structure of the inner boundary layer flow, which interferes with computation progress of fluid flows with solid walls. The present work is aiming for formulating a generalized Blasius's equation within the valid Prandtl's boundary layer by using a generalized similarity transformation and building the shear law of velocity in the inner boundary layer (velocity boundary layer, simply) both theoretically, and numerically.

    Let the leading edge of the plate flat be atx=0, the plate be parallel to thex-axis and infinitely long downstream. For simplicity, we here consider a steady flow with the main stream or outer flow velocityU∞, which is parallel to the flat plate. The most widely used boundary layer thickness is simply referred to the distancey=δ(x) from the solid boundary at which the local value of the velocity reaches 0.99 ofU∞. The scale factors foruandyappear quite naturally as the main stream velocityU∞and the boundary layer thicknessδ(x), respectively.

    1 Estimation of the boundary thickness of the leading edge point

    It is worth noting here that there are two representative thickness formulations, the displacement thicknessδ1and the momentum thicknessδ2in the Prandtl’s boundary layer theory[1]:

    (3)

    and

    (4)

    The consistent numerical estimation of the boundary layer has to work out two difficulties. First, the boundary layer approximation of the Blasius’s equation is not valid near the leading edge of the plate (δ>x) because the estimation of Eqs. (3) and (4) are not accurate. Second, according to the definition of the boundary layer thickness, the Prandtl’s boundary layer equations are only valid in the boundary layer satisfying 0≤y≤δwith the following actual boundary conditions

    (5)

    That is to say, in the inner region of boundary layer, the Prandtl’s boundary layer equations need to be transformed into an appropriate ODE as the Blasius’s equation, which should be dependent on Reynolds number.

    In the following, we suppose that the boundary layer thicknessδ(x) is the function of positionx. As well known, one can deal with a singularity similar to the that of moving contact line where the Navier-Stokes equations break down near the stagnation point with a few molecular free pathγ0≈0. Letδ0=δ(γ0) be the boundary thickness of the leading edge point, we will estimateδ0≈1/Reufor largeReuin the following. For each position x in the plate flat, one can get the exact boundary layer thickness as defined by

    numerically, wherecis a displacement(or momentum) thickness coefficient,νis the viscous coefficient, a predicted exponent. According to the Prandtl’s boundary layer theory[1], the principle of similarity of the velocity profiles in the boundary layer can be written asu/U∞=φ(y/δ), where the functionφmust be the same for a given distance x from the leading edge. Actually, experience shows that, the velocity is 0 for a suddenly accelerated plate at the stagnation point.

    Tab.1 The functionf′(η) for the boundary layer along a flat plate at zero incidence based on Eq.(1).

    ηf'Hf'L0.00.00000000.00000000.10.03222100.10050120.20.06444090.20098840.30.09665150.30136560.40.12884010.40160420.50.16098910.50204400.60.19307780.6013393ηf'Hf'L0.70.22507800.70124570.80.25695230.79842810.90.28867210.89313981.00.32019080.99000001.10.3112793??5.00.9915702

    Substituting the Blasius’s boundary conditions of Eq. (2) or actual boundary conditions of Eq. (5) into the Blasius’s Eq. (1), we get the values off′, the similar solutions to those of Howarth[4], and both solutions are listed in Table 1. However, there are some physical differences between two solutions. Howwarth’s results show that in the case of Blasius’s boundary conditions (Eq. (2)), in the actual boundaryη=1.0,f′(1.0)=0.320 190 8, it is not consistent with the definition of the Prandtl’s boundary layer thickness. In fact, the boundary conditions of Eq. (2) do not involve the viscosity. This means that the solutions of the Blasius’s equation are independent of the Reynolds number, thus the whole flow pattern in the boundary layer simply undergoes a similarity transformation, longitudinal distances and velocities remaining unchanged. Therefore, it is difficult to investigate the feature of the fluid flow with different Reynolds numbers based on these solutions. In addition, atη=5.0,f′(η)=0.99, which can not be used as the boundary conditions for the outlet flow. Even though we have used actual boundary conditions (5), the Blasius's Eq. (1) is not dependent on Reynolds number and streamline coordination, their solutions are also independent of the Reynolds number. From the view point of fluid dynamics, it is desirable to get a generalized similarity transformation which involves the viscosity.

    ( 6)

    wherec=1.720 8isthedisplacementthicknesscoefficientorc=0.664themomentumthicknesscoefficient,respectively.Approximatinglinearly, δ0≈γ0,andformulation(6)isestimatedasthemolecularfreepathnearthestagnationpointfortheflowsoflargeReynoldsnumber.ItiseasytoshowthatiftheunitReynoldsnumberReu→∞,thenδ0→0.Thereforetheclassicalboundarylayerthicknessisjustalimitoftheaboveboundarylayerthickness.

    2 The generalized Blasius's equation

    (7a)

    (7b)

    Using the dimensionless stream functionf(η), the equation of continuity (7a) can be integrated by introducing the following stream function

    Thus the velocity components become

    (8a)

    (8b)

    where the prime denoting differentiation with respect toη. Inserting Eqs. (8) into Eq. (6), we have

    (9)

    In order to obtain the generalized Blasius's equation in the boundary layer, some suitable assumptions are necessary. In order to disclose some relations between the high-Reynolds-number flows and turbulent flow from (9), we take a generalized Blasius's equation as follows

    (10)

    where fluctuation function

    The above equation is derived out with fluctuation, but the third term of the left side includes more physical meaning like surface tension, energy dissipation, et al.. In the case ofF=0, the generalized Blasius's equation is degenerated into the Blasius's Eq. (1). That is to say, the Blasius’s equation can describe inviscid supersonic flow. IfF=+∞, the Navier Stokes equations are degenerated into the reaction-diffusion equations, and the boundary layer becomes the heat diffusion boundary layer.

    To check the change of the thickness with different Reynolds number, we plot the numerical value ofδ0,δ(1) andf″ forF=0.1,0.01,0.001 and 0. As shown in Table 2, the decrease of the fluctuation function suggests an increasedf″.

    Tab.2 The change of the displacement thickness δ0 and δ(1) with different values of fluctuation function, and the boundary condition f″

    In order to analyze the feature of the boundary layer numerically, the generalized Blasius's Eq. (10) is solved with the real boundary conditions. According to the definition of the original boundary layer thickness,η≥1.0 denotes the outlet flow region of the classical boundary layer, thenf′(1)=0.99, and we get the real boundary conditions as

    (11)

    Fig.1 Shear law of the boundary layer velocity for different values of fluctuation function

    We now focus on the velocity profile in the boundary layer with various values of fluctuation function using the new boundary conditions of Eq. (11).

    Fig.1 shows the velocity profile of boundary layer. To achieve a clean velocity profile of boundary layer, a coordinates change is used for u/U in x streamline. It is seen that the Newtonian linear shear rate is perfectly satisfied for large values of fluctuation function,F=0.1 and 0.01, and the nonlinear shear law is presented for the smaller value of fluctuation function,F=0.001, or for the Blasius’s equation (corresponding toF=0). In order to show the figure clearly, the values off′ forF=0.1 and 0.01 are substituted byf′(η-0.02) andf′(η-0.01), respectively; the valuesf′ forF=0.001,or Blasius’s equation (corresponding toF=0) are substituted by that off′(η+0.02) andf′(η+0.01), respectively.

    To day, it has been well known that a boundary layer flow can switch to turbulence with incresing Reynolds number. According to the Blasius’s work as shown in Fig. 7.7 of [1]and Fig. 2(b) of [15], the velocity distribution is always nonlinear in the boundary layer for higher Reynolds number. The Blasius’s equation is independent with Reynolds number, its numerical results can not explain this feature. Our numerical results show that, with the fluctuation function decreasing, the Newtonian linear shear rate is replaced by the smaller values of fluctuation function, and there is a critical value of fluctuation function where the flow undergoes a transition toward a turbulent state, and to some extent verifying the self sustaining mechanism through destabilization of the streamwise streaks.

    明確中國(guó)有構(gòu)建國(guó)際商事法庭的必要性之后,在設(shè)計(jì)國(guó)際商事法庭的具體規(guī)則之前,我們需要考慮的是中國(guó)國(guó)際商事法庭的定位問(wèn)題,或者說(shuō)是中國(guó)國(guó)際商事法庭的管轄標(biāo)準(zhǔn)問(wèn)題。該問(wèn)題直接決定了中國(guó)國(guó)際商事法庭如何建設(shè)。

    3 the Limit Value Theorem on the velocity boundary layer

    Moreover, the Limit Value Theorem on the velocity boundary layer[17]can be straightforwardly built. By using the extended domain of definition, 0<η<3, the solution of the generalized Blasius's Eq. (10) can be expended into Taylor’s series as follows

    (12)

    Based on the boundary conditions of Eq. (10) (for a given Reynolds number,f″(0) is also given by TABLE 2, 0.9

    Sinceηwas arbitrarily chosen, the only way in which the above equation can be satisfied for all coefficients to be zero. Then

    Combining above values with Eq. (12), we have

    Then

    Letτ0=10, 0.9

    4 Conclusions and summary

    In summary, in the solution of generalized Blasius's equations, we have seen that the new boundary equation (5) is more suitable for the computation of flat plate flow. Furthermore, we show that a transition occurs from laminar to turbulent between fluctuation functionF=0.01 andF=0.001. This transition is associated with the separation point and the nonlinear shear law for different values of fluctuation function. Further, the Limit Value Theorem on the velocity boundary layer[17]is built. In conjunction with experimental measurements, the Limit Value Theorem can help to understand the mechanisms of the transition to turbulence.

    The present study can be supported by a large amount of classical experiment data published in the literature[13,15-20]. It is noted that Eq. (10) is obtained by the pressure gradient of the main stream. At point of separation,μ(?2u/?y2)|y=0=dp/dx, which is in accordance with Eq. (10) of reference[15], letF=τ/Reuwhich decreases with increasingReu. Specially, one can validate the nonlinear shear law in Fig.5 of reference[13]. Recently, Zhou et al. declared that a linear fitting is made toux(z,t) close the plate[18-24], which is opposite with our nonlinear law of boundary layer velocity of higher Reynolds number. In fact, one can find thatux(z,t)is actually nonlinear close the plate in their work (Figs 1(b) of reference[17]), and there exists an extreme value point in the boundary layer.

    [1]SCHLICHTING H, GERSTEN K. Boundary Layer Theory[M]. New York: Springer-Verlag, 2004.

    [2]YAMAGUCHI H. Engineering fluid mechanics[M]. New York: Springer-Verlag, 2009.

    [3]MEIER G E A, SREENIVASAN K R. One hundred years of boundary layer research. IUTAM Symposium[M]. Berlin: Springer-Verlag, 2004.

    [4]HOWARTH L. On the solution of the lamilar boundary-layer equations[J]. Proc R Soc London, Ser A, 1951,5:49-62.

    [5]SCHROLL R D, JOSSERAND C, ZALESKI S, et al. Impact of a viscous liquid drop[J]. Phys Rev Lett, 2009,104(3):338-346.

    [6]SIROVICH L, KARLSSON S. Turbulent drag reduction by passive mechanisms[J]. Nature, 1997,388:753-755.

    [7]CIEPLAK M, KOPLIK J, BANAVAR J R. Boundary conditions at a fluid-solid interface[J]. Phys Rev Lett, 2001,86(5):803-806.

    [8]CHOI C H, KIM C J. Large slip of aqueous liquid flow over a nanoengineered superhydrophobic surface[J]. Phys Rev Lett, 2006,96(6):66001-66100.

    [9]STEINHEUER J. Die Losungen der Blasiuss chen Grenzschicht differential gleichung[J]. Proc Wiss Ges, 1968,20:96-125.

    [10]WALEFFE F. On a self-sustaining process in shear flows[J]. Phys Fluids, 1997,9(4):883-900.

    [11]DURIEZ T, AIDER J L, WESFREID J E. Self-sustaining process through streak generation in a flat-plate boundary layer[J]. Phys Rev Lett, 2009,103(14):144502.

    [12]POPE S B. Turbulent Flows[M]. Cambridge: Cambridge University Press, 2000.

    [13]VINOD N, GOVINGARAJAN R. Pattern of breakdown of laminar flow into turbulent spots[J]. Phys Rev Lett, 2004,93(11):114501.

    [14]SEDOV L I. Continuum mechanics (Chinese, translation from the Russian)[M]. Beijing: Higher Education Press, 2009.

    [15]REEUWIJK M V, HARM J J, HANJALIC K. Wind and boundary layers in Rayleigh-Benard convection[J]. Phys Rev E, 2008,77(3):257-260.

    [16]REPIK E U. Turbulent to laminar transition of a boundary layer under large negative pressure gradients[J]. J Eng Phys, 1973,24(2):196-200.

    [17]ZHOU Q, XIA K Q. Measured instantaneous viscous boundary layer in turbulent Rayleigh-Benard convection[J]. Phys Rev Lett, 2010,104(10):104301.

    [18]QIU X L, XIA K Q. Viscous boundary layers at the sidewall of a convection cell[J]. Phys Rev E, 1998,58(1):486-491.

    [19]LAM S, SHANG X D, ZHOU S Q, et al. Prandtl number dependence of the viscous boundary layer and the Reynolds numbers in Rayleigh-Benard convection[J]. Phys Rev E, 2002,65(2):066306.

    [20]SOLOMON T H, GOLLUB J P. Sheared boundary layers in turbulent Rayleigh-Benard convection[J]. Phys Rev Lett, 1990,64(20):2382-2385.

    [21]AITCHISON I J, AO P, THOULESS D J, et al. Effective Lagrangians for BCS superconductors atT=0[J]. Phys Rev Lett B, 1994,51(10):6531-6535.

    [22]VERZICCO R, CAMUSSI R. Structure function exponents and probability density function of the velocity difference in turbulence[J]. Phys Fluids, 2002,14(2):906-909.

    [23]XU X , BAJAJ K M, AHLERS G. Heat Transport in Turbulent Rayleigh-Benard Convection [J]. Phys Rev Lett, 2000,84(19):4357-4360.

    [24]ZHONG J Q, STEVENS R, CLERCX H, et al. Prandtl-, Rayleigh-, and Rossby-Number Dependence of Heat Transport in Turbulent Rotating Rayleigh-Benard Convection[J]. Phys Rev Lett, 2009,102:044502.

    1673-5862(2016)04-0419-07

    速度邊界層剪切定律

    李明軍, 蔡振宇

    (沈陽(yáng)師范大學(xué) 數(shù)學(xué)與系統(tǒng)科學(xué)學(xué)院, 沈陽(yáng) 110034)

    自20C初,路德維?!て绽侍靥岢鱿嚓P(guān)理論以來(lái),邊界層理論被人們所熟知。然而,邊界層方程的解并不能恰當(dāng)?shù)孛枋龈呃字Z數(shù)流體。通過(guò)普朗特邊界層方程的平均值和脈動(dòng)值推導(dǎo)出帶有脈動(dòng)函數(shù)F的廣義的Blasius方程,并且通過(guò)理論推導(dǎo)和數(shù)值模擬建立內(nèi)邊界層的速度剪切定律。前緣處邊界厚度δ0=c2/Reu,其中Reu=U∞/ν,當(dāng)求位移厚度時(shí),c=1.720 8,求動(dòng)量損失厚度時(shí),c=0.664。此外,速度邊界層上的極值定理和數(shù)值實(shí)驗(yàn)表明速度邊界層的牛頓線(xiàn)性剪切定律完全滿(mǎn)足于F=0.1 和F=0.01,對(duì)于非線(xiàn)性剪切定律滿(mǎn)足于F=0.001和F→0。這樣的機(jī)制在傳統(tǒng)的邊界層理論中從未被討論過(guò)。

    廣義的Blasius方程; 波動(dòng)函數(shù); 邊界厚度; 牛頓線(xiàn)性剪切定律

    計(jì)算數(shù)學(xué)

    O242 Document code: A

    10.3969/ j.issn.1673-5862.2016.04.008

    Received date: 2016-06-20.

    Supported: Supported by National Natural Science Foundation of China(11171281).

    Biography: LI Mingjun(1968-), male, was born in Yiyang city of Hunan Province,professor of Shenyang Normal University, doctor, doctor supervisor of Xiangtan University.

    猜你喜歡
    邊界層商事定律
    基于HIFiRE-2超燃發(fā)動(dòng)機(jī)內(nèi)流道的激波邊界層干擾分析
    多一盎司定律和多一圈定律
    歡迎登錄中國(guó)商事仲裁網(wǎng)
    仲裁研究(2019年3期)2019-07-24 07:38:54
    倒霉定律
    論國(guó)際民事訴訟中《國(guó)際商事合同通則》的明示選擇適用
    萬(wàn)有引力定律
    一類(lèi)具有邊界層性質(zhì)的二次奇攝動(dòng)邊值問(wèn)題
    公司資本制度改革與商事登記制度——登記的考察日本商事
    商事法論集(2015年2期)2015-06-27 01:19:22
    商事信托的新發(fā)展與法律應(yīng)對(duì)
    商事法論集(2015年1期)2015-06-27 01:17:12
    非特征邊界的MHD方程的邊界層
    成熟少妇高潮喷水视频| 九草在线视频观看| 亚洲欧美中文字幕日韩二区| 日韩中字成人| 国产精品一区二区在线观看99 | 成人二区视频| 免费搜索国产男女视频| 我要看日韩黄色一级片| 亚州av有码| 日韩欧美 国产精品| 白带黄色成豆腐渣| 又黄又爽又刺激的免费视频.| 久久精品国产亚洲网站| 久99久视频精品免费| 最近中文字幕高清免费大全6| 嫩草影院精品99| 国产精品,欧美在线| 男人舔奶头视频| 插逼视频在线观看| kizo精华| 中文在线观看免费www的网站| 国产一级毛片在线| 国产精品嫩草影院av在线观看| 免费搜索国产男女视频| 免费不卡的大黄色大毛片视频在线观看 | 欧美性感艳星| 国产伦精品一区二区三区视频9| 国产精品一区二区三区四区久久| 成人三级黄色视频| 91av网一区二区| 欧美精品一区二区大全| 欧美最新免费一区二区三区| 波多野结衣巨乳人妻| 精品久久久久久久久久久久久| 精品无人区乱码1区二区| 午夜老司机福利剧场| 一级毛片电影观看 | 亚洲丝袜综合中文字幕| 极品教师在线视频| 国产成人aa在线观看| 亚洲高清免费不卡视频| 日本免费a在线| 又粗又硬又长又爽又黄的视频 | 久久人人爽人人爽人人片va| 国产精品久久久久久精品电影| 亚洲精品乱码久久久久久按摩| av福利片在线观看| 国产午夜精品论理片| 国产久久久一区二区三区| 能在线免费观看的黄片| 高清毛片免费观看视频网站| 欧美zozozo另类| 日韩av在线大香蕉| 日韩成人av中文字幕在线观看| 超碰av人人做人人爽久久| 五月玫瑰六月丁香| 免费不卡的大黄色大毛片视频在线观看 | 天堂√8在线中文| 搡女人真爽免费视频火全软件| 国产一区二区在线观看日韩| 亚洲自拍偷在线| 精华霜和精华液先用哪个| 亚洲成人精品中文字幕电影| av天堂在线播放| 亚洲精品自拍成人| 久久久久久九九精品二区国产| 在线天堂最新版资源| 精品一区二区三区人妻视频| 22中文网久久字幕| 国产成人精品久久久久久| 日本成人三级电影网站| 小蜜桃在线观看免费完整版高清| 99热这里只有是精品50| 欧美变态另类bdsm刘玥| kizo精华| 九草在线视频观看| 欧美激情久久久久久爽电影| 久久亚洲国产成人精品v| 日韩视频在线欧美| 欧美最黄视频在线播放免费| 亚洲在线自拍视频| 日韩大尺度精品在线看网址| 色尼玛亚洲综合影院| 亚洲久久久久久中文字幕| 狂野欧美白嫩少妇大欣赏| 免费观看的影片在线观看| 毛片一级片免费看久久久久| 日日摸夜夜添夜夜爱| 99国产精品一区二区蜜桃av| av卡一久久| 一区二区三区四区激情视频 | 久久精品国产亚洲av天美| 免费黄网站久久成人精品| 国产单亲对白刺激| 成年免费大片在线观看| 热99re8久久精品国产| 久久久色成人| 亚洲va在线va天堂va国产| 国产精品久久久久久av不卡| 亚洲欧美日韩卡通动漫| 国产精品爽爽va在线观看网站| 亚洲国产欧美在线一区| 99热网站在线观看| 国产不卡一卡二| 午夜福利在线在线| 亚洲精品乱码久久久久久按摩| 两个人的视频大全免费| 国产老妇女一区| 黄色一级大片看看| 国产伦在线观看视频一区| 国产在线男女| 亚洲丝袜综合中文字幕| 99热这里只有是精品50| 国产色爽女视频免费观看| 亚洲精品乱码久久久v下载方式| 女人被狂操c到高潮| 麻豆精品久久久久久蜜桃| 蜜桃久久精品国产亚洲av| 亚洲精品色激情综合| 3wmmmm亚洲av在线观看| 婷婷六月久久综合丁香| 一本—道久久a久久精品蜜桃钙片 精品乱码久久久久久99久播 | 99精品在免费线老司机午夜| 男女那种视频在线观看| 国产大屁股一区二区在线视频| 中文字幕久久专区| 尾随美女入室| 日韩国内少妇激情av| 网址你懂的国产日韩在线| 国产白丝娇喘喷水9色精品| 成人特级黄色片久久久久久久| 你懂的网址亚洲精品在线观看 | 岛国毛片在线播放| 网址你懂的国产日韩在线| 天堂中文最新版在线下载 | 十八禁国产超污无遮挡网站| 中文字幕久久专区| 国产伦精品一区二区三区四那| 国产伦精品一区二区三区四那| 久久久久免费精品人妻一区二区| 小说图片视频综合网站| 全区人妻精品视频| 久久精品国产自在天天线| 日韩中字成人| 日韩欧美三级三区| 国产黄a三级三级三级人| 18禁在线播放成人免费| 毛片一级片免费看久久久久| 久久午夜亚洲精品久久| 99热全是精品| 国产乱人偷精品视频| 久久久久久久亚洲中文字幕| 免费一级毛片在线播放高清视频| 日韩在线高清观看一区二区三区| 国产高清不卡午夜福利| 亚洲色图av天堂| 亚洲av二区三区四区| 国产三级中文精品| 黄片无遮挡物在线观看| 婷婷六月久久综合丁香| 国产一区二区三区在线臀色熟女| 免费搜索国产男女视频| 午夜福利视频1000在线观看| 亚洲精品影视一区二区三区av| 丝袜美腿在线中文| 国产成人a∨麻豆精品| 能在线免费看毛片的网站| 日本一二三区视频观看| 亚洲四区av| 精品无人区乱码1区二区| 晚上一个人看的免费电影| 国产乱人偷精品视频| 波多野结衣巨乳人妻| 亚洲在线自拍视频| 日韩欧美在线乱码| 欧美激情国产日韩精品一区| 国产高清不卡午夜福利| 国产极品精品免费视频能看的| 欧美一区二区精品小视频在线| 国产精品国产高清国产av| 久久久久国产网址| 日本免费一区二区三区高清不卡| 91av网一区二区| 亚洲自偷自拍三级| 日日干狠狠操夜夜爽| 国产亚洲av嫩草精品影院| 人妻少妇偷人精品九色| 人妻系列 视频| 国产精品伦人一区二区| 99久久人妻综合| 黄色欧美视频在线观看| 97超碰精品成人国产| 两性午夜刺激爽爽歪歪视频在线观看| 亚洲最大成人中文| 免费无遮挡裸体视频| 寂寞人妻少妇视频99o| 日韩精品青青久久久久久| 国产一区二区三区av在线 | 免费搜索国产男女视频| 亚洲四区av| 国产日本99.免费观看| 性欧美人与动物交配| 亚洲图色成人| 免费一级毛片在线播放高清视频| 国产伦一二天堂av在线观看| 波野结衣二区三区在线| 我的老师免费观看完整版| 国产极品天堂在线| 欧美bdsm另类| 中文精品一卡2卡3卡4更新| 免费看美女性在线毛片视频| 欧美性猛交╳xxx乱大交人| 天天躁日日操中文字幕| 欧美xxxx性猛交bbbb| 亚洲人成网站在线播| av在线老鸭窝| 91狼人影院| av免费观看日本| 亚洲精品日韩av片在线观看| 国产亚洲91精品色在线| 亚洲av一区综合| 中国美女看黄片| 欧美丝袜亚洲另类| 亚洲一区高清亚洲精品| 日韩大尺度精品在线看网址| 看免费成人av毛片| 人妻夜夜爽99麻豆av| 男人舔女人下体高潮全视频| 国产精品人妻久久久影院| 欧美性猛交╳xxx乱大交人| 男人的好看免费观看在线视频| 91精品国产九色| 国产激情偷乱视频一区二区| 国产白丝娇喘喷水9色精品| 此物有八面人人有两片| 精品午夜福利在线看| 六月丁香七月| 嘟嘟电影网在线观看| 麻豆一二三区av精品| 亚洲电影在线观看av| 久久精品国产自在天天线| 噜噜噜噜噜久久久久久91| 亚洲国产色片| 国产精品久久电影中文字幕| 国产精品不卡视频一区二区| 久久欧美精品欧美久久欧美| 99久久中文字幕三级久久日本| 成人无遮挡网站| 国产精品伦人一区二区| 精品久久久久久久人妻蜜臀av| 黄色配什么色好看| 亚洲精品自拍成人| a级毛片a级免费在线| 欧美激情在线99| 成人性生交大片免费视频hd| 亚洲不卡免费看| 色综合色国产| 日产精品乱码卡一卡2卡三| 亚洲国产欧美在线一区| 春色校园在线视频观看| 天堂影院成人在线观看| 亚洲国产高清在线一区二区三| 精品国产三级普通话版| 欧美日韩乱码在线| 国产精品久久久久久精品电影小说 | 十八禁国产超污无遮挡网站| 欧美激情国产日韩精品一区| 国内精品美女久久久久久| 国产一区二区在线观看日韩| 淫秽高清视频在线观看| 非洲黑人性xxxx精品又粗又长| av在线老鸭窝| 国产精品一区二区在线观看99 | 亚洲av免费在线观看| 日本爱情动作片www.在线观看| 99在线视频只有这里精品首页| 亚洲人成网站在线播| 人人妻人人看人人澡| 美女脱内裤让男人舔精品视频 | 一个人免费在线观看电影| 国产老妇伦熟女老妇高清| 亚洲精品乱码久久久v下载方式| 久久久欧美国产精品| 亚洲第一区二区三区不卡| 看非洲黑人一级黄片| 国产一区二区三区在线臀色熟女| 久久国内精品自在自线图片| 亚洲久久久久久中文字幕| 国产午夜精品论理片| 两个人的视频大全免费| 精品一区二区免费观看| 嫩草影院新地址| 中文字幕精品亚洲无线码一区| 国产日韩欧美在线精品| .国产精品久久| 日本色播在线视频| 国产一区二区三区在线臀色熟女| 国产老妇伦熟女老妇高清| 直男gayav资源| 国产成人freesex在线| 亚洲真实伦在线观看| 欧美成人a在线观看| 黄色欧美视频在线观看| 蜜臀久久99精品久久宅男| 亚洲精品影视一区二区三区av| 欧美成人免费av一区二区三区| 亚洲成av人片在线播放无| 91久久精品电影网| 亚洲av成人av| 国产亚洲av嫩草精品影院| 国产乱人偷精品视频| 看十八女毛片水多多多| av国产免费在线观看| 午夜亚洲福利在线播放| 欧美色欧美亚洲另类二区| 国产av麻豆久久久久久久| 国产一级毛片七仙女欲春2| 国产精品一及| 成人无遮挡网站| 国产乱人偷精品视频| 亚洲经典国产精华液单| 在线观看一区二区三区| 最近中文字幕高清免费大全6| a级毛色黄片| kizo精华| 国产亚洲5aaaaa淫片| 国国产精品蜜臀av免费| 熟女人妻精品中文字幕| 国产在线男女| 男女下面进入的视频免费午夜| 欧美成人a在线观看| 欧美成人免费av一区二区三区| 亚洲欧美日韩东京热| 国内揄拍国产精品人妻在线| 在线国产一区二区在线| 中文字幕av成人在线电影| 亚洲aⅴ乱码一区二区在线播放| 日韩大尺度精品在线看网址| 久久人人精品亚洲av| 国产精品久久视频播放| 啦啦啦韩国在线观看视频| 欧美bdsm另类| 国产精品久久久久久精品电影小说 | 黑人高潮一二区| 免费看av在线观看网站| 成年女人看的毛片在线观看| 我要搜黄色片| 成人美女网站在线观看视频| 男人舔女人下体高潮全视频| 热99re8久久精品国产| 国产老妇女一区| 一级毛片久久久久久久久女| 久久精品国产亚洲av涩爱 | 欧美区成人在线视频| 天天躁日日操中文字幕| 在线观看免费视频日本深夜| 国产亚洲av嫩草精品影院| 成人特级黄色片久久久久久久| 亚洲va在线va天堂va国产| 欧美区成人在线视频| 能在线免费看毛片的网站| 欧美日韩综合久久久久久| 亚洲婷婷狠狠爱综合网| 亚洲中文字幕日韩| 一本一本综合久久| 日韩人妻高清精品专区| 在线免费观看不下载黄p国产| 国产爱豆传媒在线观看| 人妻系列 视频| 免费看a级黄色片| 狂野欧美白嫩少妇大欣赏| 欧美三级亚洲精品| 国产淫片久久久久久久久| 12—13女人毛片做爰片一| av女优亚洲男人天堂| 国产女主播在线喷水免费视频网站 | 国产精品人妻久久久影院| 久久久久久久亚洲中文字幕| 长腿黑丝高跟| 欧美变态另类bdsm刘玥| 嫩草影院精品99| 久久久久免费精品人妻一区二区| 国产黄片视频在线免费观看| 亚洲av二区三区四区| 99国产精品一区二区蜜桃av| 在线观看66精品国产| 日韩高清综合在线| 悠悠久久av| 亚洲成人久久爱视频| 欧美+亚洲+日韩+国产| 麻豆乱淫一区二区| 婷婷色综合大香蕉| 高清在线视频一区二区三区 | 淫秽高清视频在线观看| 国产免费男女视频| 毛片一级片免费看久久久久| 亚洲电影在线观看av| 欧美色视频一区免费| ponron亚洲| 少妇人妻精品综合一区二区 | 成人性生交大片免费视频hd| 亚洲精品久久久久久婷婷小说 | 日韩欧美国产在线观看| 国产精品人妻久久久影院| 一级毛片aaaaaa免费看小| 丰满乱子伦码专区| 中文资源天堂在线| 欧美在线一区亚洲| 成人特级黄色片久久久久久久| 国产精品嫩草影院av在线观看| ponron亚洲| 国产日本99.免费观看| 亚洲欧美日韩无卡精品| 国产久久久一区二区三区| 久久久久久伊人网av| 成人美女网站在线观看视频| 成人特级av手机在线观看| 亚洲成av人片在线播放无| 少妇裸体淫交视频免费看高清| 黄色配什么色好看| 亚洲欧美清纯卡通| 午夜福利高清视频| 日本黄色片子视频| 搡女人真爽免费视频火全软件| 中文字幕人妻熟人妻熟丝袜美| 五月玫瑰六月丁香| 日韩在线高清观看一区二区三区| 亚洲av熟女| 一级黄色大片毛片| 国产精品久久久久久精品电影小说 | www日本黄色视频网| 亚洲精品乱码久久久v下载方式| 国产精品,欧美在线| 亚洲三级黄色毛片| 免费看美女性在线毛片视频| 亚洲av二区三区四区| 国产亚洲av嫩草精品影院| 国产爱豆传媒在线观看| 国产单亲对白刺激| 国产精品av视频在线免费观看| 国产又黄又爽又无遮挡在线| 国国产精品蜜臀av免费| 国产精品乱码一区二三区的特点| 在线观看午夜福利视频| 国内少妇人妻偷人精品xxx网站| 国产成人a区在线观看| 色综合亚洲欧美另类图片| 午夜精品一区二区三区免费看| 久久精品综合一区二区三区| 国产免费男女视频| 一级毛片我不卡| 国产高潮美女av| 久久99热6这里只有精品| avwww免费| 久久人妻av系列| 男女边吃奶边做爰视频| 天天一区二区日本电影三级| 性色avwww在线观看| 99久久久亚洲精品蜜臀av| 国产亚洲欧美98| 日韩精品有码人妻一区| 天堂网av新在线| 激情 狠狠 欧美| 少妇高潮的动态图| 亚洲欧美日韩高清专用| 欧美日韩在线观看h| 亚洲国产精品久久男人天堂| 99在线人妻在线中文字幕| 色5月婷婷丁香| 亚洲婷婷狠狠爱综合网| 日本免费一区二区三区高清不卡| 如何舔出高潮| 亚洲精品色激情综合| 亚洲精华国产精华液的使用体验 | 身体一侧抽搐| 久久精品久久久久久久性| 搡女人真爽免费视频火全软件| 亚洲欧美日韩高清在线视频| 青青草视频在线视频观看| 在线观看66精品国产| 成人av在线播放网站| 国产精品无大码| 午夜久久久久精精品| 国产国拍精品亚洲av在线观看| 人人妻人人看人人澡| 日韩av在线大香蕉| 国产免费一级a男人的天堂| 久久99热6这里只有精品| 九九久久精品国产亚洲av麻豆| 啦啦啦韩国在线观看视频| 天美传媒精品一区二区| 成年女人看的毛片在线观看| 色视频www国产| 免费av毛片视频| 美女被艹到高潮喷水动态| 嫩草影院新地址| 亚洲国产高清在线一区二区三| 久久人人爽人人片av| 国产高清激情床上av| 欧美潮喷喷水| 一区二区三区四区激情视频 | 欧美3d第一页| 久久久久久久久大av| 少妇的逼水好多| 99久久无色码亚洲精品果冻| 国产精品综合久久久久久久免费| 成熟少妇高潮喷水视频| 国产精品一区二区在线观看99 | 国产视频首页在线观看| 网址你懂的国产日韩在线| 成人av在线播放网站| 日韩中字成人| 欧美潮喷喷水| 日韩av在线大香蕉| 国产爱豆传媒在线观看| 国产精品久久电影中文字幕| 男女边吃奶边做爰视频| 免费无遮挡裸体视频| 人妻少妇偷人精品九色| 亚洲欧洲国产日韩| 麻豆一二三区av精品| 夜夜爽天天搞| 日韩精品青青久久久久久| 久久精品国产亚洲av天美| 97在线视频观看| 国产黄片美女视频| 一进一出抽搐动态| 男女做爰动态图高潮gif福利片| 国产伦精品一区二区三区四那| 99国产精品一区二区蜜桃av| 欧美一区二区精品小视频在线| 亚洲欧美精品综合久久99| 国产高清视频在线观看网站| 国产成人freesex在线| 老司机影院成人| 18禁黄网站禁片免费观看直播| 黄色欧美视频在线观看| 久久6这里有精品| 男的添女的下面高潮视频| 成年av动漫网址| 久久久久久久午夜电影| 一级黄色大片毛片| 亚洲成a人片在线一区二区| 久久精品国产清高在天天线| 久久亚洲国产成人精品v| 最近手机中文字幕大全| 亚洲成人精品中文字幕电影| 国产精品麻豆人妻色哟哟久久 | 成年女人永久免费观看视频| 人人妻人人澡人人爽人人夜夜 | 一级毛片久久久久久久久女| 永久网站在线| 午夜福利成人在线免费观看| 国国产精品蜜臀av免费| 日本成人三级电影网站| 日韩一区二区三区影片| 爱豆传媒免费全集在线观看| 免费看光身美女| 麻豆成人午夜福利视频| 久久这里只有精品中国| 国产精品不卡视频一区二区| 久久国内精品自在自线图片| 五月伊人婷婷丁香| 免费一级毛片在线播放高清视频| 久久午夜福利片| 国产亚洲5aaaaa淫片| 久久久久久久亚洲中文字幕| 日本爱情动作片www.在线观看| 边亲边吃奶的免费视频| 别揉我奶头 嗯啊视频| 乱人视频在线观看| 永久网站在线| 亚洲人成网站在线播| 日韩强制内射视频| 成人特级黄色片久久久久久久| 国产一级毛片在线| 精品久久久久久久久av| 一级黄色大片毛片| 久久久国产成人精品二区| 国产成人精品婷婷| 亚洲av二区三区四区| 亚洲成a人片在线一区二区| 国产高清视频在线观看网站| 伦理电影大哥的女人| 99热全是精品| 国产高清视频在线观看网站| 老熟妇乱子伦视频在线观看| 久久久久性生活片| 亚洲国产欧洲综合997久久,| 看十八女毛片水多多多| 久久99热6这里只有精品| 又粗又硬又长又爽又黄的视频 | 国产黄片视频在线免费观看| ponron亚洲| 日韩欧美国产在线观看| 69人妻影院| 日韩一本色道免费dvd| 国产亚洲av片在线观看秒播厂 | 精品国产三级普通话版| 久久亚洲国产成人精品v| 好男人在线观看高清免费视频| 草草在线视频免费看| 日韩成人伦理影院| 在现免费观看毛片| 久久精品国产亚洲av香蕉五月| 国产伦在线观看视频一区| 成人性生交大片免费视频hd| 91麻豆精品激情在线观看国产| 国产精品乱码一区二三区的特点| 最近的中文字幕免费完整| 日韩欧美三级三区| 伦理电影大哥的女人| 成年免费大片在线观看| 国产一区二区在线av高清观看| 欧美日韩综合久久久久久|