3rd Lecture : Integral Equations 1 3rd Lecture : Integral Equations Boundary Layer Theory Dept. of Naval Architecture and Ocean Engineering

Review : Governing Equations 2 Category Compressible Flow Incompressible Flow Condition Mach number Ma=U/c0>0.3 Mach number Ma=U/c0<0.3 Simplification Gov. Equations Continuity Equation Momentum Equation Energy Equation Equation of State Equations vs. Unknowns 6 Equations : Continuity 1, Momentum 3, Energy 1, Equation of State 1 6 Unknowns : ρ, u, v, w, p, T 5 Equations : Continuity 1, Momentum 3, Energy 1 5 Unknowns : u, v, w, p, T Coupling All equations are coupled, should be solved simultaneously. Energy equations is not coupled. Obtain Velocity  Solve for T (T is passive scalar).

Integral Momentum Equation (I) 3 Objectives From Boundary Layer Assumption, we know that Cf(x), δ(x), qw(x) are the functions of x only. (Actually, we can relate these quantities with some integral in y-direction.) Apply Reynolds’ transport system to CV which fits to the boundary layer  Obtain Cf(x), δ(x), qw(x) … Rate of change of internal energy inside an arbitrary CV Mass conservation For a CV with finite dimension H (H> δ) in y and infinitesimal dimension dx in x, (Mass efflux at upper & right surface) – (Mass influx at lower & left surface) = 0 : mass injection at the porous wall : mass efflux at y=H

Integral Momentum Equation (II) 4 Momentum conservation (momentum efflux) - (momentum influx) = (pressure force) + (friction force) Arranging equation using and If we introduce displacement thickness δ* and momentum thickness θ, defined as

Integral Momentum Equation (III) 5 Momentum conservation : cont’d If we let ρ=ρw, then we have Finally, rearranging and dividing both sides with Ue2 gives

Integral Momentum Equation (IV) 6 Displacement thickness Velocity defect : Integrand Ue-u  Difference between actual velocity profile and inviscid profile Adjusted inviscid profile (same mass flow rate with viscous profile) is possible only if the solid surface were displaced upward a distance δ* such that In Fig. 2-2, (Area I) = (Area II) Application : Flow rate in duct flow = (Decreased area by δ*)  (Uniform ‘core’ velocity) Momentum thickness Distance by which the surface should be shifted away from the wall to maintain momentum flux with inviscid profile.

Solution of Integral Momentum Equation (I) 7 Integral Momentum Equation : Three unknowns : If we know the velocity profile, u=u(y) then equation can be solved. How about assuming a ‘suitable shape’ ? The profiles are ‘similar’, i.e. only the nondimensional shape is required. Similar Profile shape : linear Ignorant, but bold assumption : linear profile Actually, even this simplest one satisfies no slip condition and u=Ue at y=δ !! For flat-plate boundary layer (Ue=const.) with vw=0, the IME becomes Linear shape gives Therefore, Comparison with exact solution : Surprising, because ReX-1/2 dependence is predicted !!

Solution of Integral Momentum Equation (II) 8 The Pohlhausen method : 4th order polynomial Coefficient a, b, c, d, e : Applying five boundary conditions According to the value of λ, Accelerating (Favorable gradient) λ>0 Flat Plate : λ=0 Decelerating (Adverse gradient) : λ<0 Separation : λ=-12 Pohlhausen’s pressure gradient parameter

Solution of Integral Momentum Equation (III) 9 The Pohlhausen method : cont’d From profile : can be described in terms of λ  functions of δ(x) Momentum Integral Equation again becomes an O.D.E. for δ(x). Generally applicable to non-zero pressure gradient flows, but calculations are very troublesome.  Thwaites-Walz Method

Solution of Integral Momentum Equation (IV) 10 The Thwaites-Walz method : Using empirical data Recall, for without suction/blowing Reformulation : multiply both sides with and using (shape factor) Shape factor and Shear correlation function are function of profile shape alone. If we introduce , then Rearranging equation Curve-fitting of existing experimental data gives

Solution of Integral Momentum Equation (V) 11 The Thwaites-Walz method : cont’d Multiply both sides by and rearranging Applicable to all low-speed, steady, laminar b. l. Solution procedure Invicsid solution  Ue(x) Solve for θ(x)  Λ Obtain S(Λ), H(Λ) from Table 2.1 Ex) Flat plate b.l. Invicsid solution : Ue(x) = Ue = const. Solve for θ(x) : , Λ = 0 S(Λ)=0.22 

Solution of Integral Momentum Equation (VI) 12 The Thwaites-Walz method : cont’d Alternative expression for H(Λ), S(Λ)

Integral Energy Equation (I) 13 Physics Similarly to boundary layer, there is thermal boundary layer with thickness δT(x). The ratio between δT(x) and δ(x) is governed by a physical property named Prandtl number We know that Large viscosity  “fast” momentum diffusion across y  “thin” b.l. thickness for given ΔU Similarly, Hence, (not exactly) Values of Prandtl number for various fluids Water (Liquid) : Pr ~ 10  δT ~ 3δH Air (Gas) : Pr ~ 0.7  δT ~ 1.2δH Liquid Metal : Pr ~ 10-3  δT ~ 0.03 δH

Integral Energy Equation (II) 14 Energy conservation (thermal energy efflux) - (thermal energy influx) = (frictional heating) + (heat transfer at the surface) Arranging equation

Solution of Integral Energy Equation (I) 15 Method Similarly as Integral Momentum Equation : polynomial velocity/temperature profile Unheated starting length problem Solid flat plate, Heat transfer from x=x0 Cubic polynomials for velocity/temperature profiles

Solution of Integral Energy Equation (II) 16 Unheated starting length problem : cont’d Substituting velocity/temperature profile into integral energy equation Since , we can neglect term For cubic velocity profile, integral momentum equation gives Integral energy equation can be integrated to obtain Initial condition :

Solution of Integral Energy Equation (III) 17 Unheated starting length problem : cont’d Obtain heat flux Heat transfer coefficient : a measure of heat transfer rate Dimensionless number for heat transfer : Nusselt number Another dimensionless heat transfer quantity : Stanton number We can find for x0=0, Reynolds analogy : (Heat transfer coefficient rate) (skin friction coefficiet) Prandtl number : scaling factor