Boundary layer flows The logarithmic law of the wall Mixing length model for turbulent viscosity

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1 Boundary layer flows The logarithmic law of the wall Mixing length model for turbulent viscosity Tobias Knopp D 23. November 28

2 Reynolds averaged Navier-Stokes equations Consider the RANS equations with the eddy-viscosity hypothesis by Boussinesq ( U U) = 1 P ρ + ( 2 (ν + ν t ) S( U) ) U = Lagrangian theory of turbulent mixing and the Langevin equation motivate that ν t = u 2 l turb u 2 : root mean square (r.m.s.) fluctuation velocity l turb : correlation length of the turbulence First goal: Find operational formula for ν t for generic flow situations (boundary layer, free shear layer,...)

3 obias Knopp Boundary layer flows fully developed channel flow fully developed flat-plate boundary layer flow... is a generic geometry for flow over an airfoil... y z x y=2h flow y= y z x flow

4 Turbulent channel flow This leads us to the following boundary value problem for U(y) and u v ( d u dy v ν du ) dy = u2 τ H U y= =, = 1 dp ρ dx du dy y=h = u v y= =, u v y=h = Note: The constant right hand side term given is determined by the externally prescribed pressure gradient The pressure gradient may be simply imposed by the pressure drop between channel inlet and channel outlet and the length of the channel Since there are more unknown quantities than equations, this problem is unclosed.

5 Turbulent channel flow Illustration of matching of inner layer and outer layer solution and existance of an overlap region the inner solution satisfies the b.c. at y =, but not at y = H the outer solution satisfies the b.c. at y = H, but not at y = there exists an overlap region, where inner and outer solution coincide y + = y u τ ν η = y/h overlap region y + oo η

6 Turbulent channel flow Using scaling arguments, we have derived the following forms for U Outer layer : Inner layer : du dy du dy = u τ df H dη = u2 τ ν df dy + For sufficiently high Re τ, the limits y + and η can be taken simultaneously. This implies the existence of a region of overlap or a matched layer. u τ df H dη = u2 τ ν On multiplication by y/u τ this becomes or y df H dη η df dη = yu τ ν df dy + df dy + = y + df dy + = 1 κ

7 Turbulent channel flow Viscous sublayer (y + 5) u + = y + In the inertial sublayer or log-layer or overlap-region (y + 4, η.2) F (η) = 1 log(η) + const if η 1 κ f (y + ) = 1 κ log(y + ) + const if 1 y + u v = u 2 τ if y + 1

8 Structure of the turbulent boundary layer 3 law of the wake 2 1 viscous sublayer u + = y + log layer u + = 1/κ log(y + ) + C 1 1 1

9 Ludwig Prandtl (1925) proposed Mixing length model u v = ν t U y using an algebraic model for the so-called turbulent viscosity ν t ν t = ν t ( U/ y, y) Substitution into the equilibrium stress balance equation gives nonlinear diffusion-type problem in wall normal direction ( (ν + ν t ) U ) = y y In the near-wall region: the dominant transport process of momentum is due to diffusion in wall-normal direction. Two sources of wall-normal diffusion, viz., laminar diffusion due to the thermal random motion turbulent diffusion due to the turbulent fluctuations

10 Mixing length model The Langevin equation motivates that ν t = u 2 l turb We have seen that in the log-layer u v uτ 2. = Ansatz u 2 = u τ l turb : correlation length of the turbulence. = Ansatz l turb = κy u v = ν t U y, ν t = u τ κy Note that due to the no-slip condition u v = at the wall.

11 Mixing length model Integration of the equilibrium stress balance equation from y = to y = y gives y ( (ν + ν t ) U ) dy = (ν + ν t ) U y y y y ν U y y= as ν t = u τ κy vanishes at the wall y = ; and hence (ν + ν t ) U y y = ν U y y= τ w ρ

12 Using universal coordinates Mixing length model u + = U, y + = yu τ u τ ν, ν+ t = ν t ν Then the chain rule for differentiation gives (ν + ν t ) U u + y + u + y + y (ν + ν t ) u2 τ u + ν y + (ν + ν t ) u + ν = τ w ρ = τ w ρ y + = 1 τ w uτ 2 ρ (1 + ν + t ) u+ y + = 1 In universal coordinates, Prandtl s model for ν t becomes ν t + = ν t ν = κu τ y = κy + ν

13 Mixing length model Derivation of logarithmic law of the wall (1 + ν + t ) u+ y + = 1 u + y + = ν t + u + y + = κy + 1 κy + u + = 1 κ ln(y + ) + C Model needs modification in the viscous sublayer to ensure u + = y + in the outer part of the boundary layer (law-of-the-wake region) in particular: proper transition to inviscid free-stream flow First algebraic turbulence model by Cebeci & Smith (1968)

14 Prandlt s friction law for smooth pipes Historically, for pipe flow drag is expressed in tems of the so-called friction factor D p f = L 1 2 ρu2 bulk p is the drop in pressure over an axial distance L, D is the diameter of the pipe u bulk is the bulk velocity Relation between friction factor f and skin friction coefficient c f flow f = 4c f for pipe

15 Prandlt s friction law for smooth pipes Since ln(x)dx = x ln(x) x we obtain where r + = r u τ /ν. The relation f = 4c f u bulk = 1 r U(y)dy r = u τ κr = u τ κr = ν κr = ν r r + r + log ( yuτ ) + κcdy ν ( log(y + ) + κc ) ν u τ dy + ( log(y + ) + κc ) dy + [ y + log(y + ) y + + κcy +] r κr = ν ( r + κr log(r + ) r + + κcr + ) gives us u bulk u τ = 2 c f = 8 f +

16 Prandlt s friction law for smooth pipes u bulk u τ 8 f 8 f 8 f 8 f = 1 ν ( r + κ u τ r log(r + ) r + + κcr + ) = 1 ( ( r u ) ) τ log 1 + κc κ ν = 1 ( ( ) ) ru u bulk u τ log 1 + κc κ ν u bulk ( ( = 1 ) ) f log Re 1 + κc κ 8 ( ( = 1 ) ) f log Re 1 + κc κ 8 or 1 f = 1 ( 2 2κ log Re ) f (3 + 5 log 2 2κB) 4 2κ

17 Prandlt s friction law for smooth pipes Aim: Functional dependence for the friction factor f as a function of bulk Reynolds number Re Hagen-Poiseuille friction law for laminar flow: f = 64/Re Fully developed turbulent pipe flow: Prandtl s friction law

18 Algebraic models for boundary layer flows. No-slip condition at the wall implies u v = u v and hence ν t are much smaller for y+ 1 then predicted by the Prandtl relation. Effect called blocking: Mixing in the direction normal to the wall is suppressed because the wall is impermeable and viscous v y 2 decreases much slower than u y. This effect is modelled by reducing ν t very close to the wall Van Driest: ν t = [ ( )] κy 1. e y + 2 /26 U y Using a Taylor series expansion for the van Driest damping function «u v du 2 «= ν t dy = κ2 y 2 1 e y+ /26 du 2 κ 2 y y +! 2 dy 26 + O((y + ) 2 ) y 4

19 Algebraic models for boundary layer flows Clauser 1956: proper form of the eddy viscosity in the defect layer. Idea: application of mixing length concept also to wake flows (originally also by Prandtl, 1925) ν t = αu e δ U e is the velocity at the edge of the layer and δ is called the displacement thickness and α is a closure coefficient. displacement and momentum thickness of a boundary layer: the total flux of mass and momentum is reduced due to the presence of the wall by the amount θu 2 = δ U = U(y) (U U(y)) dy (U U(y))dy

20 Algebraic models for boundary layer flows Corrsin and Kistler (1954) and Klebanoff (1954) corollary result of their experimental studies on intermittency. approaching the freestream from within the boundary layer, the flow is sometimes laminar and sometimes turbulent, i.e., it is intermittent. The eddy viscosity should be multiplied by [ ( y ) ] 6 F Kleb (y, δ 99 ) = δ

21 Cebeci-Smith model (1967) The Cebeci-Smith model is a two-layer model { ν t,in : y y m ν t = ν t,out : y y m where y m is the smallest value of y for which ν t,in = ν t,out. [ ( U ) 2 ( ) ] 2 1/2 V ν t,in = lm 2 +, l m = κy (1 e y + /A +) y x with ν t,out = αu e δ F Kleb (y, δ) ( κ =.4, α =.168, A + = y 1 ρuτ 2 ) 1/2 dp dx U e is the boundary layer edge velocity δ v is the displacement thickness

22 Cebeci-Smith model (1967) ν t,in = l 2 m [ ( U y ν t,out = αu e δ F Kleb (y, δ) ) 2 ( ) ] 2 1/2 V +, l m = κy (1 e y + /A +) x Using this model in a computed code requires a data-structure which permits to, starting at each wall node, to assess all nodes on the ray in wall normal direction, at least inside the boundary layer. Computation of δ For each node on the ray, we need to access the corresponding first node above the wall since u τ is computed there. Typically, mathcing point will lie at y + m 42, well in the log-layer.

DAY 19: Boundary Layer

DAY 19: Boundary Layer flat plate : let us neglect the shape of the leading edge for now flat plate boundary layer: in blue we highlight the region of the flow where velocity is influenced by the presence