《燃烧流体力学》（英文版）Chapter 2 Turbulent Flow

煤量甏圆家重点实验宝 Chapter 2 Turbulent Flow 8 1 Basic Concept of Turbulence 1. Classification wall turbulence- friction of wall free turbulence -interaction of fluid layers homogeneous turbulence ∫h isotropic turbulence National Lab, of Coal Combustion. HUST Wuhan 430074 P R CHINA E-mail:pcbnlcc@hust.edu.cn

NL CC National Lab. of Coal Combustion, HUST Wuhan 430074, P. R. CHINA E-mail: pcbnlcc@hust.edu.cn Chapter 2 Turbulent Flow §1 Basic Concept of Turbulence 1. Classification wall turbulence - friction of wall free turbulence - interaction of fluid layers homogeneous turbulence isotropic turbulence

煤烧圆家重点实验宝 2. Turbulence average 1)time average 0 T→> 2-1) 2)space average ()=lim-(x,10)d 2-2) X→0 X National Lab, of Coal Combustion. HUST Wuhan 430074 P R CHINA E-mail:pcbnlcc@hust.edu.cn

NL CC National Lab. of Coal Combustion, HUST Wuhan 430074, P. R. CHINA E-mail: pcbnlcc@hust.edu.cn 2. Turbulence average 1) time average  → = = T T t x t dt T x 0 0 0 ( , ) 1  ( ) lim  (2-1) 2) space average  → = x x s x t dx x t 0 0 0 ( , ) 1  ( ) lim  (2-2)

煤烧圆家重点实验宝 3)Probability density Function (x2)=「叭(x4)P()dp(2-3) 4)Reynolds average for turbulent quantity p 三-y whereφ— transient value φ— average value National Lab, of Coal Combustion. HUST pulsing value Wuhan 430074 P R CHINA E-mail:pcbnlcc@hust.edu.cn

NL CC National Lab. of Coal Combustion, HUST Wuhan 430074, P. R. CHINA E-mail: pcbnlcc@hust.edu.cn 3) Probability Density Function   − e (x0 ,t 0 ) = (x0 ,t 0 )P()d 4) Reynolds average for turbulent quantity     − ' where  ⎯ transient value  ⎯ average value ’⎯ pulsing value (2-4) (2-3)

煤量甏圆家重点实验宝 =(如-)=p-=025 Assume A=A+A' b=b+B A+b=A+a++b=atB AB=AB=0 AB=(A+A B+B)=AB+A'B OA OAOA aA + axax a National Lab, of Coal Combustion. HUST Wuhan 430074P R CHINA E-mail:pcbnlcc@hust.edu.cn

NL CC National Lab. of Coal Combustion, HUST Wuhan 430074, P. R. CHINA E-mail: pcbnlcc@hust.edu.cn  = ( − ) = − = 0 (2-5) Assume x A x A x A x A AB A A B B AB A B AB AB A B A A B B A B B B B A A A   =   +   =   = + + = + = = + = + + + = + = + = + ' ( ')( ') ' ' ' ' 0 ' ' '

煤烧圆家重点实验宝 In general,0+'an Correlation quantity of turbulence The above formulae indicate that the average of linear calculation of turbulent quantity equals linear calculation of corresponding average of turbulent values, the average of non-linear calculation of turbulent quantity equals the sum of non-linear calculation of corresponding average of turbulent values and pulsing correlation of these quantities National Lab, of Coal Combustion. HUST Wuhan 430074 P R CHINA E-mail:pcbnlcc@hust.edu.cn

NL CC National Lab. of Coal Combustion, HUST Wuhan 430074, P. R. CHINA E-mail: pcbnlcc@hust.edu.cn In general, 0 ' B' A Correlation quantity of turbulence The above formulae indicate that the average of linear calculation of turbulent quantity equals linear calculation of corresponding average of turbulent values, the average of non-linear calculation of turbulent quantity equals the sum of non-linear calculation of corresponding average of turbulent values and pulsing correlation of these quantities

煤量甏圆家重点实验宝 3. Reynolds time average equations with Reynolds average and assume PO=o po=po 2-6) )=(F。) National Lab, of Coal Combustion. HUST Wuhan 430074 P R CHINA E-mail:pcbnlcc@hust.edu.cn

NL CC National Lab. of Coal Combustion, HUST Wuhan 430074, P. R. CHINA E-mail: pcbnlcc@hust.edu.cn 3. Reynolds time average equations with Reynolds average and assume: ) ( ) i i x x  =     = =        ( ' 0  (2-6)

煤烧圆家重点实验宝 (p)+-(m,小)=-( )-(pv)+S (2-7 p=1,v1,y,h S,=0 ap +g+-(-),W O National Lab, of Coal Combustion. HUST Wuhan 430074 P R CHINA E-mail:pcbnlcc@hust.edu.cn

NL CC National Lab. of Coal Combustion, HUST Wuhan 430074, P. R. CHINA E-mail: pcbnlcc@hust.edu.cn           v S x x x v t x j j j j j j +   −     =   +   ( ) ( ) ( ) ( ') ' (2-7) s r i j j i i i s w q x v x g x p S v Y h − −     + +   = − = 0, ( ), , 1, , ,    

煤量甏圆家重点实验宝 Note: 1)in equation(2-7), the variables are averaged values 2)the new term comes from the Reynolds average of non-linear convection term termed turbulent transport flux of o. Just because of this term the number of un known variables is more than the independent equations National Lab, of Coal Combustion. HUST Wuhan 430074 P R CHINA E-mail:pcbnlcc@hust.edu.cn

NL CC National Lab. of Coal Combustion, HUST Wuhan 430074, P. R. CHINA E-mail: pcbnlcc@hust.edu.cn Note: 1) in equation (2-7), the variables are averaged values 2) the new term comes from the Reynolds average of non-linear convection term, termed turbulent transport flux of φ. Just because of this term, the number of un￾known variables is more than the independent equations

煤量烧圆家重点实验宝 §2 TurbulentⅤ scouse Coefficient Models 1. Concept of turbulent viscousity For two-dimensional boundary layer Boussinesq( 1877) proposed au t=-p=1 t -Reynolds stress u- fluid velocity in main stream ut- turbulent viscosity coefficient Wuhan 430074 P.R. CHINA E-mail:pcbnlcc@hust.edu.cn

NL CC National Lab. of Coal Combustion, HUST Wuhan 430074, P. R. CHINA E-mail: pcbnlcc@hust.edu.cn §2 Turbulent Viscouse Coefficient Models 1. Concept of turbulent viscousity For two-dimensional boundary layer, Boussinnesq(1877) proposed y u u v t t    = − ' ' =  (2-8) t - Reynolds stress u - fluid velocity in main stream t - turbulent viscosity coefficient

煤量甏圆家重点实验宝 In general Ov OVv=u, )-m水O Ox 3 p=()(=Y,h o ax Where k- time averaged value of turbulent kinetic energy of unit volume of fluid (2) 2-9) National Lab, of Coal Combustion. HUST Wuhan 430074 P R CHINA E-mail:pcbnlcc@hust.edu.cn

NL CC National Lab. of Coal Combustion, HUST Wuhan 430074, P. R. CHINA E-mail: pcbnlcc@hust.edu.cn In general, ' ' ( ) 3 2 ( ) ' ' j t j i j j i i j j i t x v k x v x v v v   − = −   +   − =           ( Y ,h)   = s Where k - time averaged value of turbulent kinetic energy of unit volume of fluid ( ) 2 1 '2 k  vi (2-9)