8th Lecture : Turbulence (I) 1 8th Lecture : Turbulence (I) Boundary Layer Theory Dept. of Naval Architecture and Ocean Engineering

Contents Basic Concept of Turbulence Mean Flow Analysis : Flat Plate 2 Basic Concept of Turbulence Characteristics of Turbulence Mean Flow Analysis : Flat Plate Multi-layer postulate Defect law Law of the wall Law of the wake Skin friction law Effect of roughness

Basic Concept of Turbulence (I) 3 Characteristics of Turbulence (I) Fluctuations in flow variable (velocity, pressure, temperature…) Three-dimensional : 2-D turbulence can’t sustain. Cf) typhoon Eddies (lumps of swirling fluid) or fluid packets of various size. Kolmogorov length scale (=0.05mm) < Eddy size <  (=40mm) The length scale covers several order of magnitudes. Mixing : active, random eddy motion  much larger mixing than laminar mixing  Skin friction & heat transfer is greatly enhanced. Self-sustaining : Turbulent flow can maintain itself by producing new eddies, to compensate viscous dissipation to heat.

Basic Concept of Turbulence (II) 4 Characteristics of Turbulence (II) Random : flow variable has a particular form of energy spectrum  neither deterministic (laminar, multi-body dynamics) nor completely random (gas dynamics). Coherent structure : largest scale motion of turbulent flow often can be considered as deterministic. Deterministic n-body dynamics Nth-order differential Equation DOF : n Completely random molecular dynamics Statistical mechanics DOF :  Turbulence

Basic Concept of Turbulence (III) 5 Characteristics of Turbulence (III) Time-averaged property Usually, overbar represents time-average operator. Reynolds’ decomposition : (instantaneous) = (time average) + (fluctuations) Turbulence intensity : ratio of RMS of velocity fluctuations to time-averaged velocity

Basic Concept of Turbulence (IV) 6 Characteristics of Turbulence (IV) Steady turbulent flow ? Turbulence is unsteady in nature. Steady (stationary) vs. Unsteady (unstationary) : time-average basis  Unsteady turbulence is the situation where the boundary condition, time average is changing with time.

Objectives of Turbulence Research 7 Recall, at the first lecture, the objective of fluid mechanics is… Determine fluid forces acting on a body surface in a moving stream ex) Drag of ship/airplane  Determine engine power Pressure drop in pipeline  Determine pumping power Heat/Mass transfer (mainly convective) For skin friction estimation of a ship… Fluctuating components of w is not so important. Therefore, in this course we are only interested in… Time-averaged (or mean) quantities : steady or unsteady, 2-D or 3-D. Cf. Fluctuating component is always unsteady and 3-D, and spans several orders of magnitude of scales. Fluctuations are always unimportant in engineering aspect ? No ! ombustion mechanics : reaction rate depends on , not on .

Mean Flow Analysis : Flat Plate (I) 8 Difference between laminar and turbulent velocity profile Turbulent flow : multi-layer appearance Outer region : Smaller velocity gradient Inner region : Large velocity gradient Similarity variable : was effective in laminar flow, but not in turbulent flow. In laminar flow, shape factor is always , but not in turbulent flow. We have to find a new correlating variable to get a collapsed plot.

Mean Flow Analysis : Flat Plate (II) 9 Ludwig Prandtl & Theodore von Kármán : multi-layer postulate (1933) Inner layer : viscous (molecular) shear dominates. depends not upon freestream patrameters. Outer layer : turbulent (eddy) shear dominates. depends not upon viscosity, distance from the wall. Overlap layer : both type important, profile smoothly connects two regions. Note : Conceptual model, so we don’t know yet where they are. Probably from the experimental data.

Mean Flow Analysis : Flat Plate (III) 10 Outer region : defect law by Clauser (1956) If velocity defect, defined as is considered, a factor proportional to will correlate all the curves. Velocity defect : relative velocity with respect to boundary layer edge Friction velocity : velocity scale derived from wall skin friction. Defect law :

Mean Flow Analysis : Flat Plate (IV) 11 Law of the wall (I) Defect law does not hold for inner (and overlap region) because the it is based on irrelevant velocity & length scales. Velocity scale : Length scale : non-dimensional group based on wall-normal distance (y), viscosity (), wall shear stress (w)  Reynolds number Law of the wall : At the inner region (very near to the wall) We can neglect velocity fluctuations, because the wall restricts the eddy motion  essentially no turbulence in near-wall region  Laminar sublayer (skin friction is only due to the molecular diffusion) We can further suppose that . ( ) This relation holds up to y+~7 (Fig. 7.4)

Mean Flow Analysis : Flat Plate (V) 12 Viscous Sublayer (y+<5) Buffer layer (5 y+  35) Logarithmic layer (35 y+  350) Wake region (y+ > 350) Overlap region (5 y+  350) Outer region

Mean Flow Analysis : Flat Plate (VI) 13 For overlap region, Defect law & Law of the wall should both hold. Multiplication inside a function = Addition outside a function  logarithmic function Clauser proposed A=5.6, B=-2.5, C=5.0 Using natural logarithm ( : von Kármán constant) This relation holds up to 35  y+ 350 (0.02  y/δ  0.2 )

Mean Flow Analysis : Flat Plate (VII) 14 For outer region We don’t know the functional relationship yet. Coles’ law of the wake Deviation of the velocity above the logarithmic law, when normalized with the maximum value of that deviation at the outer edge of the boundary layer, is a function of y/δ alone. Coles proposed that Law of the wake (=-B/2, B=-2.5 for flat plate, function of dp/dx)  : Coles’ wake parameter

Mean Flow Analysis : Flat Plate (VIII) 15 Wall-related region (sublayer + buffer + logarithmic) Spalding (1961) : single formula Whole region (sublayer + buffer + logarithmic + wake) Szablewski (1969)

Mean Flow Analysis : Flat Plate (IX) 16 Skin friction law From Letting y=δ, and eliminating U and y gives Implicit formula : Cf should be solved iteratively. Other correlation for skin friction coefficient Schultz-Grunow (1940) Schoenherr (1932) For total frictional resistance coefficient, CD

Effect of Pressure Gradient (I) 17 For non-zero pressure gradient flow When applying velocity defect plot  No collapse Defect law should be modified with additional parameter to be universal. Recall : Falkner-Skan solution : similarity solution exists for wedge flows (parameter β) Clauser : Equilibrium turbulent flow Clauser’s equilibrium parameter β : should be constant. Velocity-defect plot

Effect of Pressure Gradient (II) 18 Extension of law of the wake Coles’ wake parameter Then use law of the wake

Effect of Pressure Gradient (III) 19 Extension of law of the wake : cont’d Other boundaey layer parameters

Effect of Roughness (I) 20 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) 21 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.