9th Lecture : Turbulence (II) 1 9th Lecture : Turbulence (II) Boundary Layer Theory Dept. of Naval Architecture and Ocean Engineering

Contents Effect of roughness Flow in a pipe 2 Effect of roughness Flow in a pipe The central problem of the analysis of turbulent flows Mean flow turbulent transport formulations Mean flow models for the eddy viscosity and mixing length Characteristics of Turbulence

Effect of Roughness (I) 3 Effect : depends not only on size but also on pattern There is no universal theory. Effect of size (k) ‘Smooth’ roughness : size less than viscous sublayer no effect on the flow, wall can be considered ‘smooth’. k+> 10 : sublayer begins to disappear. Skin friction increases. Logarithmic portion of the smooth wall curve shifts down and right. For sand-grain roughness

Effect of Roughness (II) 4 Effect : depends not only on size but also on pattern Effect of size (k) Fully rough (k+ 70) : roughness penetrates inner layer  independent of viscosity. Defect law holds.

Flow in a pipe (I) Fully developed flow Momentum balance leads to 5 Fully developed flow Momentum balance leads to Resistance coefficient (for laminar flow, ) Blasius (1913) Prandtl (1935)

Flow in a pipe (II) 6 Effect of roughness Moody chart

Flow in a pipe (III) Comparison with flat plate 7 Comparison with flat plate Law of the wall : directly applicable to pipe flow Modified defect law :

Analysis of turbulent flows (I) 8 Governing equation Same as laminar flow : G. E. is still valid for random fluctuations. Then, what’s the difference ? In practical point of view, we are mainly interested in time-averaged variables. Reynolds’ decomposition : Time average + Fluctuations Property of time average operation Substitute Reynolds’ decomposition and taking time average for continuity Continuity : satisfied by both time-average and fluctuations.

Analysis of turbulent flows (II) 9 Reynolds’ Decomposition For momentum equation, Taking time average, Usually, we can assume Therefore, momentum equation becomes Reynolds stress

Analysis of turbulent flows (III) 10 Closure problem G. E. (again) 2 equations for 3 unknowns ( ) : can’t be solved  opened ! Closure of turbulence : We have to close the equations with additional equation. Closure problem always takes place in turbulence. For 3D, 4 equations for 11 unknowns What shall we do ?  Turbulence modeling.

Analysis of turbulent flows (IV) 11 Kinetic theory of gas Laminar shear stress : momentum transfer between molecules having different velocity (velocity gradient).  (mean free path) : average distance between collision of molecules The momentum transported to the molecule at x2=0, by the molecule at x2=- Let’s neglect higher order terms. Then, If there are N molecules/volume of the average speed of , total momentum transfer per unit time and unit area is the shear stress. For Newtonian fluid, Gas molecule

Analysis of turbulent flows (V) 12 Nature of Reynolds stress Reynolds stress is the momentum transfer between fluid particles (eddies) having different velocity (velocity gradient).

Analysis of turbulent flows (VI) 13 Modeling of Reynolds stress : Gradient transport formulation Turbulent momentum transfer is associated with the velocity gradient (Unfortunately, Not necessarily…, this is actually incorrect assumption). Eddy viscosity : Boussinesq (1877) We can think of “eddy viscosity”. Then, the G. E. becomes Note, is not a fluid (thermodynamic) property. It is “flow property”.  Function of flow variables.

Analysis of turbulent flows (VII) 14 Mixing length model : Prandtl (1925) Similarly with the mean free path in the kinetic theory of gas, we can think of “mixing length” in turbulent flow. Then, the eddy viscosity is described as  dimensionally correct. In boundary layer, Prandtl and von Kármán estimated that ( : von Kármán constant)

Analysis of turbulent flows (VIII) 15 Fallacy of “Gradient Transport Formulation” This can’t explain the turbulent momentum transfer where the velocity gradient is zero. ex) free shear layer such as jet flow However, in wall-bounded shear flow (boundary layer), it works well.

Analysis of turbulent flows (IX) 16 Derivation of Logarithmic law of the wall In the inner region, it is known that Also, in the overlap region Therefore,

Analysis of turbulent flows (X) 17 Outer region Prandtl : mixing length in outer region is proportional to boubdary layer thickness. Clauser’s eddy viscosity Buffer region + Logarithmic region van Driest (1956) model

Analysis of turbulent flows (XI) 18 Buffer region + Logarithmic region van Driest (1956) model

Mean Flow Analysis : Flat Plate (VIII) 19 Wall-related region (sublayer + buffer + logarithmic) Spalding (1961) : single formula Whole region (sublayer + buffer + logarithmic + wake) Szablewski (1969)