2 Boundary Layer Features Boundary Layers: Physical FeaturesVelocity Boundary LayerFor flows with high Reynolds number:i.e. relatively high velocity, large size,and/or low viscosity.The viscous region is confined to aboundary layer along the surface,characterized by shear stressesand velocity gradients.Boundary layer thickness :Why does increase in the flow direction?Manifested by a surface shear stressthat add up to a friction drag force, :How does vary in the flow direction? Why?
3 Boundary Layer Features (cont.) Thermal Boundary LayerLikewise temperature variationis also confined in a thermalboundary layer, characterizedby temperature gradients and heatfluxes.Thermal boundary layer thickness :increases in the flow direction.Manifested by a surface heat fluxand a convection heat transfer coefficient h.If is constant, how do andh vary in the flow direction?
4 Local and Average Coefficients Distinction between Local andAverage Heat Transfer CoefficientsLocal Heat Flux and Coefficient:Average Heat Flux and Coefficient for a Uniform Surface Temperature:For a flat plate in parallel flow:
5 Boundary Layer Transition Transition criterion for a flat plate in parallel flow:What may be said about transition if ReL < Rex,c?If ReL > Rex,c?
6 Effect of transition on boundary layer Transition (cont.)Effect of transition on boundary layerthickness and local convection coefficient:Why does transition cause a significantincrease in the boundary layer thickness?Why does the convection coefficientdecay in the laminar region?Why does it increase significantlywith transition to turbulence?Why does the convection coefficientdecay in the turbulent region?
7 Boundary Layer Equations Laminar Boundary Layer EquationsConsider concurrent velocity and thermal boundary layer development for steady,two-dimensional, incompressible flow with constant fluid propertiesand negligible body forces (X=0, Y=0). Boundary layer approximations:
8 Boundary Layer Equations (cont.) Conservation of Mass:Conservation of Momentum:Why can we express the pressure gradient as dp∞/dx instead ofConservation of Energy:
9 Similarity Considerations Boundary Layer SimilarityAs applied to the boundary layers, the principle of similarity is based ondetermining similarity parameters that facilitate application of results obtainedfor a surface experiencing one set of conditions to geometrically similar surfacesexperiencing different conditions.Dependent boundary layer variables of interest are:For a prescribed geometry, the corresponding independent variables are:Geometrical: Size (L), Location (x, y)Hydrodynamic: Velocity (V)Fluid Properties:
10 Similarity Considerations (cont.) Key similarity parameters may be inferred by non-dimensionalizing the momentumand energy equations.Recast the boundary layer equations by introducing dimensionless forms of theindependent and dependent variables.Neglecting viscous dissipation, the following normalized forms of the x-momentumand energy equations are obtained:
11 Similarity Considerations (cont.) How may the Reynolds and Prandtl numbers be interpreted physically?For a prescribed geometry,The dimensionless shear stress, or local friction coefficient, is thenWhat is the functional dependence of the average friction coefficient?
12 Similarity Considerations (cont.) For a prescribed geometry,The dimensionless local convection coefficient is thenWhat is the functional dependence of the average Nusselt number?How does the Nusselt number differ from the Biot number?
13 Reynolds AnalogyThe Reynolds AnalogyEquivalence of dimensionless momentum and energy equations fornegligible pressure gradient (dp*/dx*~0) and Pr~1:Advection termsDiffusionHence, for equivalent boundary conditions, the solutions are of the same form:
14 Reynolds Analogy (cont.) For Pr = 1, the Reynolds analogy, which relates important parameters of the velocityand thermal boundary layers, isModified Reynolds (Chilton-Colburn) Analogy for 0.6 < Pr <60:An empirical result that extends applicability of the Reynolds analogy:Colburn j factor for heat transferApplicable to laminar flow if dp*/dx* ~ 0.Generally applicable to turbulent flow without restriction on dp*/dx*.
16 Problem: Turbine Blade Scaling Problem 6.19: Determination of heat transfer rate for prescribed turbine blade operating conditions from wind tunnel data obtained for a geometrically similar but smaller blade. The blade surface area may be assumed to be directly proportional to its characteristic length