1. 3rd Lecture : Integral Equations1
3rd Lecture : Integral Equations
Boundary Layer Theory
Dept. of Naval Architecture and Ocean Engineering

2. Review : Governing Equations2
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).

3. 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

4. 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

5. 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

6. 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.

7. 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 !!

8. 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

9. 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

10. 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

11. 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 

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

13. 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

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

15. 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

16. 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 :

17. 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