# Boundary Layer Theory

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Boundary Layer Theory

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- Prof. Dr. Norbert EbelingBoundary Layer TheoryLecture notesProf. Dr. N. Ebeling Boundary Layer Theory- 1 - Contents : 1) General fluid mechanics / Newton fluids 1.1) Euler's law of hydrostatics 1.2) Friction 1.3) Dimensionless numbers 1.4) Laminar flow in a tube 2) Conservation equations 2.1) Mass balance for = const. 2.2) Euler's and Bernoulli's equations 2.3) Navier-Stokes equations 3) Boundary layers 3.1) Boundary layers on a flat plate 3.2) Friction forces on a plate 3.3) Boundary layer on an obstacle 4) Potential and stream functions 5) Law of Kutta-Joukowski 6) Exact calculation of the Boundary layer thickness 6.1) Conservation of mass (continuity equation) 6.2) Navier-Stokes and Blasius equations 6.3) Friction 7) Thermal Boundary layer 8) Mass Transfer Boundary layer equation 9) Turbulent Boundary layer10) Burbling11) Bibliography12) Acknowledgment(dz)dydx-Du = dmDtidF = 0ut-u = u xDuDtif = dx dy dz dm - - -xyzie x uje y v general definitionsk e z w`),,,Prof. Dr. N. EbelingBoundary Layer Theory - 2 - 1) General fluid mechanics / Newton Fluids

General definitions

Acceleration :

stationary :

frequently : volume force: (e.g. g ) - - -u u u u = + u + v + w t x y zDuDt1dF 2dF x F fdx dy dz = dxx- - - - - dp dy dz dF = - -xp f= x-x1 2 fdV + dF= dF - -Prof. Dr. N. Ebeling Boundary Layer Theory- 3 - 1.1) Euler's law of hydrostatics

x xf = +uy -u = y - inertial forcefrictionforce + dyy - ,,( )= dydx dzFRFydA - - -_{Prof. Dr. N. EbelingBoundary Layer Theory - 4 - 1.2) Friction Moving fluid :( Couette - flow ) Newton fluid

Schlichting : 1.3) Dimensionless numbers : Reynolds number Re ~

u dx dy dz u xRe ~ dx dy dzydm- - - - -- - -u u ~ ; = y yux d y| | | \ -u ~ y d y | | | \ -v dRe = -with = Prof. Dr. N. Ebeling Boundary Layer Theory- 5 - or any comparable speedv elselaminar flow : high friction forces,low inertial forces avoided by friction

deciding2V v v ddRe = = v d- -- --AAFC= p s -21 u2p - - ( or ) analogouswC ( )R2md dp = or dx u2 --Froude -numbervFrg d=-RwFC= p s -Prof. Dr. N. EbelingBoundary Layer Theory - 6 - ascending forces Bernoulli :

Pipe : Gravity influence :

64 = ReRr dp du = = - 2 dx dr - -22R dp ru(r) = - 14 dx R (| | ( |\ ( - -Prof. Dr. N. Ebeling Boundary Layer Theory- 7 -1.4) Laminar flow in a tube

extremely high nearly no initial forces, no influence of dm or ! Hagen-Poisseulle : Derivation : Integration with u (r = R) = 0 leads to : 2d dp 6464 = = dx v d v d v2 - --- - --2 2dpp r-p + dx r- 2 r dx = 0dx| | |\ - - - - -v + = 0yux R0V =u (r) 2dr `- -4 R dpV =- 8dx | | |\ -`--22V R pu = = R 8l `-- -( )u2RRe = - -64 = laminar !!ReProf. Dr. N. EbelingBoundary Layer Theory - 8 - 2) Conservation equationsImportant conservation equations for describing continuous flow ( cartesian coordinates ) :

2.1) Mass balance for = const.

212p d = u l-- -1yz u - -2= u yz - -2+ v xz - -( )1 2 2u- u y = + v x - -u p u = - x x - -xDu udV = dV u = + dFDt x - - - - -u v + = 0x y Prof. Dr. N. Ebeling Boundary Layer Theory- 9 - 2.2) Euler's and Bernoulli's equations Eulers equation ( one direction, pipe ):

Integration :W = F lleads to Bernoulli's equation

xu - p u = + fx x - - -22 2 21 1 1u = p- g h 2 - - -( )=dydx dzRdFy - - -u u pu + v = - x y xDuDt| | | \ - - -v Prof. Dr. N. EbelingBoundary Layer Theory - 10 - Mechanical energy balance : Bernoulli incl. hydrostatics

Euler (2 directions ): v leads to a higher value of u2.3) Navier - Stokes - equation Bernoulli and Euler neglect frictionu y =-2 2x2 2u u pu + v = f- + + x y xu uy x | || | | | \ \ - - - -2 2y2 2pv + u = f- + + v v v vy x y y x | || | | | \ \ - - - -Prof. Dr. N. Ebeling Boundary Layer Theory- 11 - Navier - Stokes - Equations ( Can be simplified in a boundary layer (later)) 3) Introduction to Boundary layers 3.1) Boundary layers on a flat plateNo influence of the viscosity but directly on the wallBoundary layer phenomena :( Schlichting )2 22 2u = + yRufx| | | \ -x RxDu p = f- + fDt x - -22u u ~ x - - x ~ u-2u u ~ ; ~ ux x y -2 22u u = = ux y y - - -99 (x) x = 5 u--Prof. Dr. N. EbelingBoundary Layer Theory - 12 -Thickness of a boundary layer, laminar on a plate inertial force = friction force ( Navier -Stokes ) ( )= f x u( )iy = 0 (x) =U - u(x,y)dyU - - value low99 is arbitrary 99 (x)5 x = lRel-Prof. Dr. N. Ebeling Boundary Layer Theory- 13 - Dimensionless :

A non - arbitrary value : displacement thickness

3.2) Friction forces on a plate :highvalue991 3i -u( ) = ywwx | | |\ -W Ww2S(Surface)F F = c= = E u b l2- -( )lW W0F= bxdx - -l 1 32W0F~ b u xdx - - - - -Prof. Dr. N. EbelingBoundary Layer Theory - 14 - xu ~ with ~ uw --3 u ~ wx - -132WF~ b u2 l - - - - -324 2b 2 ul ~ bul4wc - - - - -- - -l ~ Rewc1,1328 = RewcProf. Dr. N. Ebeling Boundary Layer Theory- 15 -

3.3) Boundary layer on an obstacle : Navier - Stokes : Far away from the obstacle (stream line) : ( )dU l dpU = -no frictiondx dx - -dU dp and are related to Bernoullidx dx =w ds -

Prof. Dr. N. EbelingBoundary Layer Theory - 16 -

4) Potential and Stream functions For describing vortex streams ( and comparable ) : Circulation : Potential streams (no friction ) : no rotation Mass balance ; conservation equation :1122v = = t x = = -t1 v u = - 2 x yuy | | | \ v u = 0 ; - = 0x y v + = 0yux u = ;v = - y x +-= 0y x x y| | | | | | \ \ 2 22 2+ = 0x y = ; v = yux Prof. Dr. N. Ebeling Boundary Layer Theory- 17 - Stream function (definition ) : Conservation equation : No rotation : Potential function : Potential streams 1 v u v u = - ; - = 02 x y x y| | | \ -2 22 2u u + = 0x y ( )p = fu, v22v u u v - - + = 0y x y x x x y| | | | | | ||| \ \ \ u u pu + v =- + 0x y x| | | \ - - -Prof. Dr. N. EbelingBoundary Layer Theory - 18 - Streams without any rotation :

also conservation equation :

Insert in Navier - Stokes :

leads to Bernoulli for v = 0 - no friction ! -no rotation - no friction v u - = 0x y u v + = 0x y 2 2v u - = 0 - = 0x y x y x y 2 22 2+ = 0x y Prof. Dr. N. Ebeling Boundary Layer Theory- 19 - Model frequently used : On the obstacle : boundary layerin the vicinity , but outside thelayer :no friction potential function : No rotation : Conservation equations : Stream function : Conservation equations o.k.

u = ; v = x y u = ; v = - y x =w ds ,,-

Prof. Dr. N. EbelingBoundary Layer Theory - 20 - from definition : Stream line :( no v : )-> = constant Circulation : Example : here : = 0( all possible ways ) airfoil :high speed low speedu + v = 0x y - - 0 ( )= 2r r - - -Potential- and flowfunctions as well as velocitys for some elementary potential flowsflow streamlinetranslational flowsource flow( productiveness E )potential vortex stream( circulation I' )source-drain flow( productiveness E, distance h )dipole flow( dipole moment M )( ) x, y ( ) x, y ( ) u x, y ( ) v x, yU x + V y E ln r2 212E r ln2 r 2M x 2 r U y - V x E 2ln r2( )1 2E - 2 2M y 2 rU2E x 2 r 2y 2 r2 21 2E x + h x - 2 r r| | | \ 2 24M y- x 2 r V2E y 2 r 2x 2 r2 21 2Ey 1 1 - 2 r r| | | \ 4M 2xy 2 rlrw ~ = 0 Prof. Dr. N. Ebeling Boundary Layer Theory- 21 - assumption : ; obviously : One exception :including the centre : (see also: Gersten, K. : Einfhrung in die Strmungsmechanik, Bertelsm. Univ.Verlag, 1st edition, page 130 )radyield : E = w2r - -for x = r : E = u 2x - -Eu = 2 x ( or r ) 2 2E = ln x+ y2-2 2 2 2E 1 1 1u = = 2xx 2 2x+ y x+ y - - - -2E xu = 2 r -E E y = =arctg2 2 x| | | \ - -E yu = =arctgy 2 y x| | | | || \ \ -radspring : V = w2r h `- - -Prof. Dr. N. EbelingBoundary Layer Theory - 22 -

Spring :

21 arctg x = x 1 + x( )( )yxyxu = x-Prof. Dr. N. Ebeling Boundary Layer Theory- 23 - Bronstein : the rest is the same For application : airfoil : ( )22 2yxE 1 1 xu = 2 x x1 + - - -2E xu = 2 r -stream =of model streams AF =blp - -( ) 2A1F =bl2u2 - - - -2AF= 2 l b u - - - -Prof. Dr. N. EbelingBoundary Layer Theory - 24 - 5) Law of Kutta - Joukowski simple example : flat plate : Kutta - Joukowski = 2 ul - -AF= b u - - - : u = U y22u u + v = x yuuy - - - = 0 : u = 0, v = 0 y uyx -Prof. Dr. N. Ebeling Boundary Layer Theory- 25 - 6) Exact calculation of the Boundary layer thickness Boundary layer on a plate : For similarity y/ (x) is important v + = 0yux ( )x vx~ u-m = = y m yu -( )= uf m u-( )1 1 =2 uf m2xx- - - - - = y 2 xum-- -( )=2 x f m2 xuu u - - - - -- -Prof. Dr. N. EbelingBoundary Layer Theory - 26 - Definition : ( the factor 2 is arbitrary but helpful ) Idea : Stream function :

= - xv( )=2 x uf m - - - -( )Schlichtingsays dimensionlessstream function =2 x u fm - - - -( )m2 x uf mx (+ ( - - - - -3-2u 1 = y-x22mx | | |\ - - --3- 1 22 u u = f -yx xx 2 x 2 x | | | | \ -- - - - -- - -( ) uv = - =m f- fx 2 x -- --( )3-21 = uf my- x22u ux | | |\ - - - - -- uv = m f- f2 x y 2 xuy y (( (( (( - -- - -- -( )1 = f m - 2 xuu mx| | |\ - - --( )2 x uf m - - - - -Prof. Dr. N. Ebeling Boundary Layer Theory- 27 - 6.1) Conservation of mass (continuity equation) u uu u u v = f+yf2 x 2 x 2 x 2 x 2 xmy - -- - - - -- - - - - - - -u v + = 0x y ( ) =m f- f2 xuv-- --( )= f m u u-( )1 = f mm-2 xuux| | |\ - - --( )u u = uf my 2 x - -- -Prof. Dr. N. EbelingBoundary Layer Theory - 28 - Conti - equation

6.2) Navier-Stokes and Blasius equations Navier-Stokes for the boundary layer on a flat plate : f2 x 2 xu u -- -- - -( )1 = m f m2 xvuy- - --( )22 =f m2 x 2 xu u uuy - - -- - - -22u u uu + v = x y y - - - f+ f f= 0Blasius - Equation -m = 0f = 0 , f= 0m : f= 1( )m=0i.e. y = 0u = 0 and f= 0( )m i.e. y u = uand f= 1 ( ) uv = m f- f2 x-- --fory = 0v and f have to be 0 Prof. Dr. N. Ebeling Boundary Layer Theory- 29 -with : Inserting and differentation leads direc