EXTERNAL INCOMPRESSIBLE VISCOUS FLOW Nazaruddin Sinaga KULIAH X EXTERNAL INCOMPRESSIBLE VISCOUS FLOW Nazaruddin Sinaga

Main Topics The Boundary-Layer Concept Boundary-Layer Thickness Laminar Flat-Plate Boundary Layer: Exact Solution Momentum Integral Equation Use of the Momentum Equation for Flow with Zero Pressure Gradient Pressure Gradients in Boundary-Layer Flow Drag Lift

The Boundary-Layer Concept

The Boundary-Layer Concept

Boundary Layer Thickness

Boundary Layer Thickness Disturbance Thickness, d where Displacement Thickness, d* Momentum Thickness, q

Boundary Layer Laws The velocity is zero at the wall (u = 0 at y = 0) The velocity is a maximum at the top of the layer (u = um at =  ) The gradient of BL is zero at the top of the layer (du/dy = 0 at y =  ) The gradient is constant at the wall (du/dy = C at y = 0) Following from (4): (d2u/dy2 = 0 at y = 0)

Navier-Stokes Equation Cartesian Coordinates Continuity X-momentum Y-momentum Z-momentum

Laminar Flat-Plate Boundary Layer: Exact Solution Governing Equations For incompresible steady 2D cases:

Laminar Flat-Plate Boundary Layer: Exact Solution Boundary Conditions

Laminar Flat-Plate Boundary Layer: Exact Solution Equations are Coupled, Nonlinear, Partial Differential Equations Blassius Solution: Transform to single, higher-order, nonlinear, ordinary differential equation

Boundary Layer Procedure Before defining and * and are there analytical solutions to the BL equations? Unfortunately, NO Blasius Similarity Solution boundary layer on a flat plate, constant edge velocity, zero external pressure gradient

Blasius Similarity Solution Blasius introduced similarity variables This reduces the BLE to This ODE can be solved using Runge-Kutta technique Result is a BL profile which holds at every station along the flat plate

Blasius Similarity Solution

Blasius Similarity Solution Boundary layer thickness can be computed by assuming that  corresponds to point where U/Ue = 0.990. At this point,  = 4.91, therefore Wall shear stress w and friction coefficient Cf,x can be directly related to Blasius solution Recall

Displacement Thickness Displacement thickness * is the imaginary increase in thickness of the wall (or body), as seen by the outer flow, and is due to the effect of a growing BL. Expression for * is based upon control volume analysis of conservation of mass Blasius profile for laminar BL can be integrated to give (1/3 of )

Momentum Thickness Momentum thickness  is another measure of boundary layer thickness. Defined as the loss of momentum flux per unit width divided by U2 due to the presence of the growing BL. Derived using CV analysis.  for Blasius solution, identical to Cf,x

Turbulent Boundary Layer Black lines: instantaneous Pink line: time-averaged Illustration of unsteadiness of a turbulent BL Comparison of laminar and turbulent BL profiles

Turbulent Boundary Layer All BL variables [U(y), , *, ] are determined empirically. One common empirical approximation for the time-averaged velocity profile is the one-seventh-power law

Results of Numerical Analysis

Momentum Integral Equation Provides Approximate Alternative to Exact (Blassius) Solution

Momentum Integral Equation Equation is used to estimate the boundary-layer thickness as a function of x: Obtain a first approximation to the freestream velocity distribution, U(x). The pressure in the boundary layer is related to the freestream velocity, U(x), using the Bernoulli equation Assume a reasonable velocity-profile shape inside the boundary layer Derive an expression for tw using the results obtained from item 2

Use of the Momentum Equation for Flow with Zero Pressure Gradient Simplify Momentum Integral Equation (Item 1) The Momentum Integral Equation becomes

Use of the Momentum Equation for Flow with Zero Pressure Gradient Laminar Flow Example: Assume a Polynomial Velocity Profile (Item 2) The wall shear stress tw is then (Item 3)

Use of the Momentum Equation for Flow with Zero Pressure Gradient Laminar Flow Results (Polynomial Velocity Profile) Compare to Exact (Blassius) results!

Use of the Momentum Equation for Flow with Zero Pressure Gradient Turbulent Flow Example: 1/7-Power Law Profile (Item 2)

Use of the Momentum Equation for Flow with Zero Pressure Gradient Turbulent Flow Results (1/7-Power Law Profile)

Pressure Gradients in Boundary-Layer Flow

DRAG AND LIFT Fluid dynamic forces are due to pressure and viscous forces acting on the body surface. Drag: component parallel to flow direction. Lift: component normal to flow direction.

Drag and Lift Lift and drag forces can be found by integrating pressure and wall-shear stress.

Drag and Lift In addition to geometry, lift FL and drag FD forces are a function of density  and velocity V. Dimensional analysis gives 2 dimensionless parameters: lift and drag coefficients. Area A can be frontal area (drag applications), planform area (wing aerodynamics), or wetted-surface area (ship hydrodynamics).

Drag Drag Coefficient with or

Drag Pure Friction Drag: Flat Plate Parallel to the Flow Pure Pressure Drag: Flat Plate Perpendicular to the Flow Friction and Pressure Drag: Flow over a Sphere and Cylinder Streamlining

Drag Flow over a Flat Plate Parallel to the Flow: Friction Drag Boundary Layer can be 100% laminar, partly laminar and partly turbulent, or essentially 100% turbulent; hence several different drag coefficients are available

Drag Flow over a Flat Plate Parallel to the Flow: Friction Drag (Continued) Laminar BL: Turbulent BL: … plus others for transitional flow

Drag Coefficient

Drag Flow over a Flat Plate Perpendicular to the Flow: Pressure Drag Drag coefficients are usually obtained empirically

Drag Flow over a Flat Plate Perpendicular to the Flow: Pressure Drag (Continued)

Drag Flow over a Sphere : Friction and Pressure Drag

Drag Flow over a Cylinder: Friction and Pressure Drag

Streamlining Used to Reduce Wake and Pressure Drag

Lift Mostly applies to Airfoils Note: Based on planform area Ap

Lift Examples: NACA 23015; NACA 662-215

Lift Induced Drag

Lift Induced Drag (Continued) Reduction in Effective Angle of Attack: Finite Wing Drag Coefficient:

Lift Induced Drag (Continued)

Fluid Dynamic Forces and Moments Ships in waves present one of the most difficult 6DOF problems. Airplane in level steady flight: drag = thrust and lift = weight.

Example: Automobile Drag Scion XB Porsche 911 CD = 1.0, A = 25 ft2, CDA = 25 ft2 CD = 0.28, A = 10 ft2, CDA = 2.8 ft2 Drag force FD=1/2V2(CDA) will be ~ 10 times larger for Scion XB Source is large CD and large projected area Power consumption P = FDV =1/2V3(CDA) for both scales with V3!

Drag and Lift For applications such as tapered wings, CL and CD may be a function of span location. For these applications, a local CL,x and CD,x are introduced and the total lift and drag is determined by integration over the span L

Friction and Pressure Drag Fluid dynamic forces are comprised of pressure and friction effects. Often useful to decompose, FD = FD,friction + FD,pressure CD = CD,friction + CD,pressure This forms the basis of ship model testing where it is assumed that CD,pressure = f(Fr) CD,friction = f(Re) Friction drag Pressure drag Friction & pressure drag

Streamlining Streamlining reduces drag by reducing FD,pressure, at the cost of increasing wetted surface area and FD,friction. Goal is to eliminate flow separation and minimize total drag FD Also improves structural acoustics since separation and vortex shedding can excite structural modes.

Streamlining

Streamlining via Active Flow Control Pneumatic controls for blowing air from slots: reduces drag, improves fuel economy for heavy trucks (Dr. Robert Englar, Georgia Tech Research Institute).

CD of Common Geometries For many geometries, total drag CD is constant for Re > 104 CD can be very dependent upon orientation of body. As a crude approximation, superposition can be used to add CD from various components of a system to obtain overall drag. However, there is no mathematical reason (e.g., linear PDE's) for the success of doing this.

CD of Common Geometries

CD of Common Geometries

CD of Common Geometries

Flat Plate Drag Drag on flat plate is solely due to friction created by laminar, transitional, and turbulent boundary layers.

Flat Plate Drag Local friction coefficient Laminar: Turbulent: Average friction coefficient For some cases, plate is long enough for turbulent flow, but not long enough to neglect laminar portion

Effect of Roughness Similar to Moody Chart for pipe flow Laminar flow unaffected by roughness Turbulent flow significantly affected: Cf can increase by 7x for a given Re

Cylinder and Sphere Drag

Cylinder and Sphere Drag Flow is strong function of Re. Wake narrows for turbulent flow since TBL (turbulent boundary layer) is more resistant to separation due to adverse pressure gradient. sep,lam ≈ 80º sep,turb ≈ 140º

Effect of Surface Roughness

Lift Lift is the net force (due to pressure and viscous forces) perpendicular to flow direction. Lift coefficient A=bc is the planform area

Computing Lift Potential-flow approximation gives accurate CL for angles of attack below stall: boundary layer can be neglected. Thin-foil theory: superposition of uniform stream and vortices on mean camber line. Java-applet panel codes available online: http://www.aa.nps.navy.mil/~jones/online_tools/panel2/ Kutta condition required at trailing edge: fixes stagnation pt at TE.

Effect of Angle of Attack Thin-foil theory shows that CL≈2 for  < stall Therefore, lift increases linearly with  Objective for most applications is to achieve maximum CL/CD ratio. CD determined from wind-tunnel or CFD (BLE or NSE). CL/CD increases (up to order 100) until stall.

Effect of Foil Shape Thickness and camber influences pressure distribution (and load distribution) and location of flow separation. Foil database compiled by Selig (UIUC) http://www.aae.uiuc.edu/m-selig/ads.html

Effect of Foil Shape Figures from NPS airfoil java applet. Color contours of pressure field Streamlines through velocity field Plot of surface pressure Camber and thickness shown to have large impact on flow field.

End Effects of Wing Tips Tip vortex created by leakage of flow from high-pressure side to low-pressure side of wing. Tip vortices from heavy aircraft persist far downstream and pose danger to light aircraft. Also sets takeoff and landing separation at busy airports.

End Effects of Wing Tips Tip effects can be reduced by attaching endplates or winglets. Trade-off between reducing induced drag and increasing friction drag. Wing-tip feathers on some birds serve the same function.

Lift Generated by Spinning Superposition of Uniform stream + Doublet + Vortex

Lift Generated by Spinning CL strongly depends on rate of rotation. The effect of rate of rotation on CD is small. Baseball, golf, soccer, tennis players utilize spin. Lift generated by rotation is called The Magnus Effect.

The End Terima kasih

Derivation of the boundary layer equations II

Blassius exact solution I Boundary layer over a flat plate = Ue1 Variable transformation: , Stream function definition: , Wall boundary condition:

Blassius exact solution II Boundary layer over a flat plate Ordinary differential equation Boundary conditions The analytical solution of the ordinary differential equation was obtained by Blasius using series expansions

Blassius exact solution III Boundary layer over a flat plate Velocity along x1-direction: Velocity along x2-direction: Boundary layer thickness: Displacement thickness: Wall shear stress:

Blassius exact solution IV Boundary layer over a flat plate

Von Karman integral momentum equation I Momentum conservation along x1-direction

Von Karman integral momentum equation II Solution Considerations: p(x1) = pe(x1)

Approximate solutions I Linear, quadratic, cubic and sinusoidal velocity profiles Assumption of a self-similar velocity profile U1*= f (x2*) Specifications of the boundary conditions 3. Resolution of the Von Karman integral momentum equation

Approximate solutions II Example: quadratic velocity profiles General form + boundary conditions = velocity profile along x1-dir Von Karman integral momentum equation solution boundary layer thickness

Approximate solutions III Example: quadratic velocity profiles Displacement thickness: from the definition => Velocity along x2-direction from continuity equation Wall shear stress from =>