火羽流第3讲-[兼容模式]

火羽流理论火羽流理论第三讲理想羽流之2第三讲理想羽流之－GaussianPlumes陆守香：sxlu@ustceducn陆守香：[e-mail]办公地点：力学一楼214室0551-3603141，13956966718FarFieldGaussianPlumes3.1引言－羽流特征参数3.2Gaussian羽流回顾-弱羽流假设回顾弱羽流假设„Assumethesimplecaseofapointsourceofheatatheightz=0ofheatatheightz=0.„Theenergyisconsideredtobetotallytransportedintheplume,andnoradiativeheatisemittedfromthepointsource.„Theforcedrivingthesystemcantherefore„Theforcedrivingthesystemcanthereforebeassumedtoariseduetothedensitydifferenceofthehotairabovethepointandthecoldsurroundingair.„Furtherassumethattheflowprofileacross„Furtherassumethattheflowprofileacrossthesectionoftheplume,atanyheight,isaso-calledtophatprofile.Therefore,theupwardvelocityisassumedtobeconstantacrossthewidthoftheplumeandzeropoutsideit.„Theplumetemperatureissimilarlyassumedtobeconstantacrossanysectionoftheplume.„Assumethatthereisarelationshipbetweentheupwardvelocityintheplumeandthehorizontalentrainmentvelocityintotheplume,v,suchthatv=α·u,whereαiscalledtheentrainmentwhereαiscalledtheentrainmentcoefficient.3.1引言－羽流特征参数引言羽流特征参数„Rouse，Yih&Humphries,Gravitationalconvectionfromaboundarysource,Tellus,1952,4:202~2102exp()urubz⎡⎤⎛⎞=−⎢⎥⎜⎟⎢⎥⎝⎠⎣⎦2()expmubzTTrβ∞⎢⎥⎝⎠⎣⎦⎡⎤⎛⎞−=−⎢⎥⎜⎟exp()mTTbzβ∞⎢⎥⎜⎟−⎢⎥⎝⎠⎣⎦,1()rbzηβ=„具有相似解3.1引言－羽流特征参数引言羽流特征参数„为了无量纲化，需要引入特征参数在点源羽流中没有源直径速度等参数如何办？„在点源羽流中，没有源直径、速度等参数，如何办？„假设存在一个长度参数Zc和一个速度参数uc，进行无量纲化回顾-2.2基本方程回顾基本方程2QubcTπρ=Δ&pcTπρ∞Δ2Q&2pQubcTπρ∞=Δ量纲相等量纲相等并定义2Q&Qu=&2ccpQuzcTρ∞∞=2ccpuzcTρ∞∞=回顾-2.4动量方程和浮力方程回顾动量方程和浮力方程由质量流率方程„由质量流率方程„由浮力方程1/51/2Q⎛⎞⎜⎟&量纲相等并定义1/2cpQugcTgρ∞∞=⎜⎟⎜⎟⎝⎠2/5Q⎛⎞⎜⎟&22cccuzgz=2cQuT=&cpQzcTgρ∞∞⎛⎞=⎜⎟⎜⎟⎝⎠2ccpzcTρ∞∞2Qgz⎛⎞=⎜⎟⎜⎟&⎝⎠2ccpgzzcTρ∞∞=⎜⎟⎜⎟⎝⎠2/5Q⎛⎞&cpQzcTgρ∞∞⎛⎞=⎜⎟⎜⎟⎝⎠pgρ∞∞⎝⎠特征长度zc是一个非常重要的参数，与火源直径一样，决定了大涡旋结构的尺寸和火焰的高度。所以通常把无量纲火焰高度表示为所以，通常把无量纲火焰高度表示为fczzfDD⎛⎞=⎜⎟⎝⎠DD⎜⎟⎝⎠3.2SolutionforGaussianprofilep假设1一个点火源模型。因此，基本尺寸D变为0。2使用Bousinnesq假设，除了浮力项，密度和其它流体性质为常数2使用Bousinnesq假设，除了浮力项，密度和其它流体性质为常数3u/um和(T-T0)/(Tm-T0)都用高斯分布2⎡⎤2exp()murubz⎡⎤⎛⎞=−⎢⎥⎜⎟⎢⎥⎝⎠⎣⎦2exp()TTrTTbzβ∞⎣⎦⎡⎤⎛⎞−=−⎢⎥⎜⎟−⎢⎥⎝⎠⎣⎦()mTTbz∞⎢⎥⎝⎠⎣⎦3.2SolutionforGaussianprofilep假设假设4、空气卷吸基于恒定的空气卷吸常数α，它是空气卷吸量与轴向速度的比值。另外，可以用动量的比值来表示空气卷吸系数，这需要引入(ρ/ρ0)1/2。4、能量释放速率被假定在燃烧区域均一分布，且直接由空气卷吸，率中的量ΔHc/sn决定，其中ΔHc是燃烧热，s是化学当量的空气燃料比，n是卷吸的空气与燃烧消耗的空气的比值。n是独立的流体动力学因子，为羽流混合的特征量。在我们的计算中ΔH/s被取为2.91kJ/g因子，为羽流混合的特征量。在我们的计算中ΔHc/s被取为2.91kJ/g，与甲烷一致。5、火焰辐射被视为恒定的分数Xr。6对于浮力控制的羽流燃料供应的质量动量和能量流率被认6、对于浮力控制的羽流，燃料供应的质量、动量和能量流率被认为很小，可以忽略。7、羽流完全是湍流。回顾-弱羽流假设回顾弱羽流假设„Assumethesimplecaseofapointsourceofheatatheightz=0ofheatatheightz=0.„Theenergyisconsideredtobetotallytransportedintheplume,andnoradiativeheatisemittedfromthepointsource.„Theforcedrivingthesystemcantherefore„Theforcedrivingthesystemcanthereforebeassumedtoariseduetothedensitydifferenceofthehotairabovethepointandthecoldsurroundingair.„Furtherassumethattheflowprofileacross„Furtherassumethattheflowprofileacrossthesectionoftheplume,atanyheight,isaso-calledtophatprofile.Therefore,theupwardvelocityisassumedtobeconstantacrossthewidthoftheplumeandzeropoutsideit.„Theplumetemperatureissimilarlyassumedtobeconstantacrossanysectionoftheplume.„Assumethatthereisarelationshipbetweentheupwardvelocityintheplumeandthehorizontalentrainmentvelocityintotheplume,v,suchthatv=α·u,whereαiscalledtheentrainmentwhereαiscalledtheentrainmentcoefficient.3.2SolutionforGaussianprofilep2⎡⎤⎛⎞2exp()murubz⎡⎤⎛⎞=−⎢⎥⎜⎟⎢⎥⎝⎠⎣⎦2exp()mTTrTTbzβ∞∞⎡⎤⎛⎞−=−⎢⎥⎜⎟−⎢⎥⎝⎠⎣⎦定义无量纲参数()/mTTT∞∞Φ=−//mccUuuBbz==/czzζ=3.2SolutionforGaussianprofilep2⎡⎤⎛⎞积分羽流方程并令2exp()murubz⎡⎤⎛⎞=−⎢⎥⎜⎟⎢⎥⎝⎠⎣⎦PTCζ=Φ积分羽流方程，并令2exp()mTTrTTbzβ∞∞⎡⎤⎛⎞−=−⎢⎥⎜⎟−⎢⎥⎝⎠⎣⎦mCBζnuUCζ=TCζΦ定义无量纲参数mlCBζ=()/mTTT∞∞Φ=−//mccUuuBbz==1.Morton,B.R.,Taylor,G.I.,andTurner,J.S.,Proc.OftheRoyalSoc.ofLondon,Ser.A234:1–23(1956).2.Lee,S.L.andEmmons,H.W.,J.ofFluidMech./czzζ=,,,f11:353–368(1961).3.Steward,F.R.,Combust.Flame8:171–178(1964).3.2SolutionforGaussianprofilep()/TTTΦ=0.098α=()//mmcTTTUuu∞∞Φ=−=2/30.91310.58(1)TrCβχ==−//ccBbzzzζ==1/34.17(1)0.118urlCCχ=−=5/3TCζ−Φ=21.64Uζ=Φ1/3TuCUCBCζζζ−Φ=2/32/32/31/34/3225(1)(1)324TrCβχπβα−⎡⎤+⎛⎞=−⎢⎥⎜⎟⎝⎠⎢⎥⎣⎦23lBCUζβ==β⎝⎠⎢⎥⎣⎦1/32251(1)Cβαχ−⎡⎤⎛⎞+⎛⎞=−⎢⎥⎜⎟⎜⎟6lCα=2βζΦ(1)24urCαχπβ⎢⎥⎜⎟⎜⎟⎝⎠⎝⎠⎣⎦5l3.2SolutionforGaussianprofilep23U0.098α=32Uβζ=Φ2/30.91310.58(1)TrCβχ==−1/34.17(1)0.118urlCCχ=−=2/mcugz21.64Uζ=Φ22()//mcTTTzzu∞∞−2()/3mmuFrgTTTz∞∞==−232rFβ=3.2SolutionforGaussianprofilep3.2SolutionforGaussianprofilep