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

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

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). +

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

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

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)

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)

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).

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)

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 3-3.3 (pp.119-121) of F. M. White For Elliptic section,

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

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

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 

Boundary Layer Concept (I) Unbelievably, d’Alembert’s paradox has remained unsolved before… Ludwig Prandtl (1875-1953) - Advisor of von Kármán - Father of Aerodynamics Theory proposed Boundary Layer Concept in 1904.

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.

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

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.

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.

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

Turbulence (II) Sketch of Leonardo da Vinci

Turbulence (III)