1. 2nd Lecture : Boundary Layer Concept
Boundary Layer Theory
Dept. of Naval Architecture and Ocean Engineering

2. Contents Review : Navier-Stokes Equation Energy Equation
Some Exact Laminar Solutions
d’Alembert’s Paradox
Boundary Layer Concept
Separation and Kutta Condition
Basics of Turbulence

3. Review : Navier-Stokes Equation
Momentum equation + Constitutive Relation
For incompressible flow, or
Solving N-S equation
4 equations (1 continuity + 3 momentum) for 4 unknowns (u, v, w, p) : Theoretically Solvable
In reality, it is extremely difficult !!!  Nonlinearity of convection term
For very limited cases, exact laminar solutions can be obtained.
Neglect the nonlinearity from simple geometry & steadiness.
Similarity Transform : P.D.E  O.D.E (will be dealt with in Lecture 3). +

4. Conservation Law : Energy (I)
Using Reynolds’ transport theorem
If we let α = ρ(e+V2) (internal energy + kinetic energy)  Energy Conservation
Rate of change of internal energy inside an arbitrary CV
Rate of energy change inside CV = Work done on CV + Heat transfer to CV
Surface integral  Volume integral (Gauss’ theorem)
Work done
by stress
Work done
by body force
Heat transferred
into CV

5. Conservation Law : Energy (II)
Derivation of equations
Energy equation becomes
With stress-strain relation,
=0 from continuity
ρuj  momentum eqn.  0
compression work
Dissipation function : mechanical energy  heat

6. Conservation Law : Energy (III)
Derivation of equations : cont’d
For incompressible flow, dissipation function is always positive (‘positive-definite’)  irreversible energy transfer from mechanical to heat.
Compression work term becomes
Therefore, the energy equation becomes
If we introduce enthalpy he+p/ρ, dh=CpdT
What is enthalpy ?
Enthalpy = Internal Energy + Mechanical Work
Convenient for calculating energy change of system during reactions in open surrounding, i. e., at constant pressure
Cp : specific heat at constant pressure
Cv : specific heat at constant volume, de= CvdT, Cp = Cv+ R for ideal gas (p=ρRT)

7. Conservation Law : Energy (IV)
Derivation of equations : cont’d
Another constitutive relation : Fourier’s heat conduction law, heat flux – temperature relation, proportionality constant k  heat conductivity
Final form of energy equation
Simplification for incompressible, constant property fluid
For incompressible flow, we can neglect compression work and C=Cv=Cp
In addition for constant property fluid (k=const.),
Thermal diffusivity vs. kinematic viscosity (momentum diffusivity)
Diffusivity = (Diffusion Coefficient)/(Inertia Coefficient)

8. Governing Equations 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).

9. Exact Solutions (I) Couette Flow : wall-driven shear flow
Flow between two infinite plate, with upper one moving with U
Steady 1-D flow, no dependence of B.C. with x-, z- direction
G.E. simplifies as
General Couette flow
For plane Couette flow (dp/dx=0)

10. Exact Solutions (II)
Poiseuille Flow : Fully-developed duct flow after entrance region
Velocity in purely axial direction
G.E. simplifies as
2D Poisson equation : analytic solutions for various cross-sectional shapes. (dp/dx=const.)
For non-circular cross section : consult Section (pp ) of F. M. White
For Elliptic section,

11. Exact Solutions (III)
Hagen-Poiseuille Flow (1840) : Circular duct (Radius R)
Resistance coefficient λ (Darcy friction factor) : pressure drop per unit length of pipe
Laminar flow up to Re D =2,300

12. d’Alembert’s Paradox (I)
Hydrodynamics : Beautiful Mathematics
Kelvin’s theorem : For ideal (incompressible, inviscid) fluids, the circulation (vorticity) will be preserved.
Flow of an ideal fluid around body inside an irrotational field in the infinity (e.g. uniform flow) will remain irrotational everywhere.  Irrotational flow
Irrotational flow   Potential flow
Flow around a body : superposition of velocity potentials to make stream surface.
d’Alembert’s paradox : Flow around arbitrary body in uniform flow
Let’s think about a CS (control surface) S0 enclosing an arbitrary body of surface S.
S is stream surface  No momentum flux across S.
Force exerted on the fluid : Momentum integral on S0

13. d’Alembert’s Paradox (II)
d’Alembert’s paradox (cont’d)
If we use velocity potentials to generate a flow around a body
After some algebra, we get
Force exerted on arbitrary body inside a uniform flow is zero.  No Drag Force at all !
What’s wrong with this ?
Unrealistic assumptions : No momentum flux across S0
In real situation,
Shear stress on S0  Skin friction drag
Flow separation  Form drag 

14. Boundary Layer Concept (I)
Unbelievably, d’Alembert’s paradox has remained unsolved before…
Ludwig Prandtl ( )
- Advisor of von Kármán
- Father of Aerodynamics Theory
proposed Boundary Layer Concept in 1904.

15. Boundary Layer Concept (II)
Momentum equation revisited
If we nondimensionalize the equation using representative scales (U, L)
We can’t neglect the diffusion (viscous) term even though it is order of 1/Re.
Prandtl’s reasoning
There would exist a confined in a thin layer near the body, where the viscous effect can’t be neglected.
Two regions;
Inner region (Boundary Layer) : velocity changes from 0 to U.  Large velocity gradient  Non-negligible diffusion term
External region : neglect diffusion term and use Euler equation.
Solve the flow separately and then combine two solution.

16. Boundary Layer Concept (III)
Consequence of boundary layer assumption
If we let δ be the thickness of b. l. (δ<<1)  dδ/dx << 1  very small streamline curvature, streamline is parallel to the surface.
Therefore, pressure is constant in wall-normal direction.
Then, we can …
use pressure p(x) in inviscid solution calculated from the Euler equation
calculate outer edge velocity Ue(x) for the boundary layer from
Combination of the two solution is in single way. or

17. Separation and Kutta Condition (I)
Boundary layer separation occurs
in adverse pressure gradient : pressure increases in streamwise direction to resist the flow.
Streamline depart from the body surfaces.
Separation is undesirable because
Lift can’t be obtained from wing.
Form drag / Pressure drop increases.
Turbulence in separation bubble.
Boundary layer assumption does not hold any more after separation.

18. Separation and Kutta Condition (II)
Kutta condition in hydrodynamics
an empirical condition to select a proper, unique inviscid solution in a sharp trailing edge.
In hydrodynamics, only normal component of velocity is controlled  Infinite number of solutions is possible with varying circulation.
Kutta condition in viscous flow
The flow should ‘turn around’ a sharp trailing edge undergoing adverse pressure gradient  separation.
The real separation point moves farther downstream, coincides with the trailing edge.
Kutta condition is to impose a realistic condition to inviscid solution at trailing edge.

19. Turbulence (I) Turbulence … changes randomly with time.
consists of lumps of swirling fluid, called eddy.
Random, complicated…

20. Turbulence (II)
Sketch of Leonardo da Vinci

21. Turbulence (III)