PHAROS UNIVERSITY ME 253 FLUID MECHANICS II Boundary Layer (Two Lectures)
Flow Past Flat Plate Dimensionless numbers involved for external flow: Re>100 dominated by inertia, Re<1 – by viscosity
Boundary Layer Flows over bodies. Examples include the flows over airfoils, ship hulls, etc. Boundary layer flow over a flat plate with no external pressure variation. U Dye streak U U U laminar turbulent transition
Boundary layer characteristics for large Reynolds number flow can be divided into boundary region where viscous effect are important and outside region where liquid can be treated as inviscid
The Boundary-Layer Concept
The Boundary-Layer Concept
Boundary Layer Thicknesses
Boundary Layer Thicknesses Disturbance Thickness, d Displacement Thickness, d* Momentum Thickness, q
Boundary Layer Thickness Boundary layer thickness is defined as the height above the surface where the velocity reaches 99% of the free stream velocity.
Boundary Layer(BL) Three Thicknesses of a Boundary Layer d
Displacement Thickness Volume flux: Ideal flux:
Velocity Distribution Ideal Fluid Flow Velocity Defect Solid Boundary * Velocity Defect Equivalent Flow Rate
Eqn. for Displacement Thickness By equating the flow rate for velocity defect to flow rate for ideal fluid If density is constant, this simplifies to * would always be smaller than
Displacement Thickness Laminar B.L.
Eqn. for Momentum Thickness By equating the momentum flux rate for velocity defect to that for ideal fluid If density is constant, this simplifies to would always be smaller than * and
Momentum Thickness The rate of mass flow across an element of the boundary layer is (r u dy) and the mass has a momentum (r u2 dy ) The same mass outside the boundary layer has the momentum (r u ue dy) Q is a measure of the reduction in momentum transport in the B. Layer
Empirical Equations of Laminar B. Layer Parameters Boundary Layer Thickness Momentum Thickness Displacement Thickness Skin Friction Coefficient
Skin Friction Coefficient
Boundary layer characteristics Boundary layer thickness Boundary layer displacement thickness: Boundary layer momentum thickness (defined in terms of momentum flux):
Drag on a Flat Plate Drag on a flat plate is related to the momentum deficit within the boundary layer Drag and shear stress can be calculated by assuming velocity profile in the boundary layer
Boundary Layer Definition Boundary layer thickness (d): Where the local velocity reaches to 99% of the free-stream velocity. u(y=d)=0.99U Displacement thickness (δ*): Certain amount of the mass has been displaced (ejected) by the presence of the boundary layer ( mass conservation). Displace the uniform flow away from the solid surface by an amount δ* Amount of fluid being displaced outward equals d* U-u
Laminar Flat-Plate Boundary Layer: Exact Solution Governing Equations
Laminar Flat-Plate Boundary Layer: Exact Solution Boundary Conditions
Laminar Flat-Plate Boundary Layer: Exact Solution Results of Numerical Analysis
MOMENTUM INTEGRAL EQN BOUNDARY LAYER EQUATIONS BOUNDARY CONDITIONS y=0, u=v=0 & y = δ , u= U Integrating the momentum equation w.r.t y in the interval [0,δ]
MOMENTUM INTEGRAL EQN
DISPLACEMENT THICKNESS MOMENTUM INTEGRAL EQN DISPLACEMENT THICKNESS MOMENTUM THICKNESS
MOMENTUM INTEGRAL EQN
MOMENTUM INTEGRAL EQN VON KARMAN MOMENTUM INTEGRAL EQUATION
KARMAN POHLHAUSEN METHOD FOR FLOW OVER FLAT PLATE BERNOULLI’S EQUATION NO IMPOSED PRESSURE VON KARMAN EQUATION REVISED KARMAN EQUATION FOR NO EXTERNAL PRESSURE
KARMAN POHLHAUSEN METHOD FOR FLOW OVER FLATE PLATE Dimensionless velocity u/U can be expressed at any location x as a function of the dimensionless distance from the wall y/δ. Boundary (at wall) conditions y=0, u=0 ; d2u/dy2 = 0 At outer edge of boundary layer (y=δ) u=U, du/dy=0 Applying boundary conditions a=0,c=0, b=3/2, d= -1/2 Velocity profile
KARMAN POHLHAUSEN METHOD FOR FLOW OVER FLATE PLATE INSERT INTO THIS EQUATION
KARMAN POHLHAUSEN METHOD FOR FLOW OVER FLATE PLATE
KARMAN POHLHAUSEN METHOD FOR FLOW OVER FLATE PLATE At the solid surface, Newton’s Law of Viscosity gives:
KARMAN POHLHAUSEN METHOD FOR FLOW OVER FLATE PLATE BLASIUS SOLUTION KARMAN POHLHAUSEN SOLUTION
Boundary Layer Parameters BOUNDARY LAYER THICKNESS INCREASES AS THE SQUARE ROOT OF THE DISTANCE x FROM THE LEADING EDGE AND INVERSELY AS SQUARE ROOT OF FREE STREAM VELOCITY. WALL SHEAR STRESS IS INVERSELY PROPORTIONAL TO THE SQUARE ROOT OF x AND DIRECTLY PROPORTIONAL TO 3/2 POWER OF U LOCAL & AVERAGE SKIN FRICTION VARY INVERSELY AS SQUARE ROOT OF BOTH x & U
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 (Blasius) 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)
Example Assume a laminar boundary layer has a velocity profile as u(y)=U(y/d) for 0yd and u=U for y>d, as shown. Determine the shear stress and the boundary layer growth as a function of the distance x measured from the leading edge of the flat plate. u=U y u(y)=U(y/d) x
Note: In general, the velocity distribution is not a straight line Note: In general, the velocity distribution is not a straight line. A laminar flat-plate boundary layer assumes a Blasius profile. The boundary layer thickness d and the wall shear stress tw behave as:
Laminar Boundary Layer Development Boundary layer growth: d x Initial growth is fast Growth rate dd/dx 1/x, decreasing downstream. Wall shear stress: tw 1/x As the boundary layer grows, the wall shear stress decreases as the velocity gradient at the wall becomes less steep.
Momentum Integral Relation for Flat-plate BL y P Free stream U Stream line d h x x CV U=const P=const Steady & incompressible
Momentum Integral Relation for Flat-plate BL Outlet Inlet Continuity
Meantime For flat plate boundary layer
Shape factor
Skin-friction coefficient Drag coefficient
Boundary Layer Equation Inviscid
3 Boundary Layer Equation
Boundary Layer Equation 2-D,steady,incompressible,neglect body force
For BL, External flow (Inviscid Flow)
Euler Equation Laminar flow Turbulent flow
Blasius 1908
DISPLACEMENT AND MOMENTUN THICKNESS Typical distribution of d, d* and q U =10m/s, n=17x10-6 m2/s d d* q
Influence of Adverse Pressure Gradient Adverse Pressure Gradient dp/dx>0 can cause flow separation
Pressure Gradients in Boundary-Layer Flow
Boundary Layer and separation Flow decelerates Flow accelerates Constant flow Flow reversal free shear layer highly unstable Separation point
Flow Separation Separation Boundary layer q Wake Inviscid curve Stagnation point 1.0 Turbulent Laminar -1.0 -2.0 -3.0 q
Drag Coefficient: CD Stokes’ Flow, Re<1 Supercritical flow turbulent B.L. Relatively constant CD
Drag Drag Coefficient with or
Influence of Adverse Pressure Gradient Adverse Pressure Gradient dp/dx>0 can cause flow separation
WHY DOES BOUNDARY LAYER SEPARATE? Adverse pressure gradient interacting with velocity profile through B.L. High speed flow near upper edge of B.L. has enough speed to keep moving through adverse pressure gradient Lower speed fluid (which has been retarded by friction) is exposed to same adverse pressure gradient is stopped and direction of flow can be reversed This reversal of flow direction causes flow to separate Turbulent B.L. more resistance to flow separation than laminar B.L. because of fuller velocity profile To help prevent flow separation we desire a turbulent B.L.
EXAMPLE OF FLOW SEPARATION Velocity profiles in a boundary layer subjected to a pressure rise (a) start of pressure rise (b) after a small pressure rise (c) after separation Flow separation from a surface (a) smooth body (b) salient edge
BOUNDARY LAYER SEPARATION Separation takes place due to excessive momentum loss near the wall in a boundary layer trying to move downstream against increasing pressure, i.e., which is called adverse pressure gradient. MOMENTUM EQUATION AT WALL v=u=0
BOUNDARY LAYER SEPARATION In adverse gradient, the second derivative of velocity is positive at the wall, yet it must be negative at the outer layer (y=δ) to merge smoothly with the mainstream flow U(x). It follows that the second derivative must pass through zero somewhere in between, at the point of inflexion, and any boundary layer profile in an adverse must exhibit a characteristic S –shape.
BOUNDARY LAYER SEPARATION In FAVOURABLE GRADIENT, profile is rounded, no point of inflexion, no separation. In ZERO PRESSURE GRADIENT, point of inflexion is at the wall itself. No separation. In ADVERSE GRADIENT, point of inflexion (PI) occurs in the boundary layer, its distance from the wall increasing with the strength of the adverse gradient CRITICAL CONDITION is reached where the wall shear is exactly zero (∂u/∂y =0). This is defined as separation point SHEAR STRESS AT WALL IS ZERO
BOUNDARY LAYER SEPARATION
BOUNDARY LAYER SEPARATION The mathematical explanation of flow-separation : The point of separation may be defined as the limit between forward and reverse flow in the layer very close to the wall, i.e., at the point of separation This means that the shear stress at the wall, .But at this point, the adverse pressure continues to exist and at the downstream of this point the flow acts in a reverse direction resulting in a back flow.
EXAMPLE: SLATS AND FLAPS