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PHAROS UNIVERSITY ME 253 FLUID MECHANICS II

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.

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PHAROS UNIVERSITY ME 253 FLUID MECHANICS II

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  1. PHAROS UNIVERSITYME 253 FLUID MECHANICS II Boundary Layer (Two Lectures)

  2. Flow Past Flat Plate • Dimensionless numbers involved • for external flow: Re>100 dominated by inertia, Re<1 – by viscosity

  3. 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 turbulent laminar transition

  4. 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

  5. The Boundary-Layer Concept

  6. The Boundary-Layer Concept

  7. Boundary Layer Thicknesses

  8. Boundary Layer Thicknesses • Disturbance Thickness, d • Displacement Thickness, d* • Momentum Thickness, q

  9. Boundary Layer Thickness Boundary layer thickness is defined as the height above the surface where the velocity reaches 99% of the free stream velocity.

  10. Boundary Layer(BL) Three Thicknesses of a Boundary Layer d

  11. Displacement Thickness Volume flux: Ideal flux:

  12. Velocity Distribution U U Velocity Defect Ideal Fluid Flow Solid Boundary * Velocity Defect Equivalent Flow Rate

  13. 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 

  14. Displacement Thickness Laminar B.L.

  15. 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 

  16. 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

  17. Empirical Equations of Laminar B. Layer Parameters Boundary Layer Thickness Momentum Thickness Displacement Thickness Skin Friction Coefficient

  18. Skin Friction Coefficient

  19. Boundary layer characteristics • Boundary layer thickness • Boundary layer displacement thickness: • Boundary layer momentum thickness (defined in terms of momentum flux):

  20. 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

  21. 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

  22. Laminar Flat-PlateBoundary Layer: Exact Solution • Governing Equations

  23. Laminar Flat-PlateBoundary Layer: Exact Solution • Boundary Conditions

  24. Laminar Flat-PlateBoundary Layer: Exact Solution • Results of Numerical Analysis

  25. 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]

  26. MOMENTUM INTEGRAL EQN

  27. MOMENTUM INTEGRAL EQN DISPLACEMENT THICKNESS MOMENTUM THICKNESS

  28. MOMENTUM INTEGRAL EQN

  29. MOMENTUM INTEGRAL EQN • VON KARMAN MOMENTUM INTEGRAL EQUATION

  30. KARMAN POHLHAUSEN METHOD FOR FLOW OVER FLAT PLATE BERNOULLI’S EQUATION NO IMPOSED PRESSURE VON KARMAN EQUATION REVISED KARMAN EQUATION FOR NO EXTERNAL PRESSURE

  31. 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

  32. KARMAN POHLHAUSEN METHOD FOR FLOW OVER FLATE PLATE INSERT INTO THIS EQUATION

  33. KARMAN POHLHAUSEN METHOD FOR FLOW OVER FLATE PLATE

  34. KARMAN POHLHAUSEN METHOD FOR FLOW OVER FLATE PLATE • At the solid surface, Newton’s Law of Viscosity gives:

  35. KARMAN POHLHAUSEN METHOD FOR FLOW OVER FLATE PLATE BLASIUS SOLUTION KARMAN POHLHAUSEN SOLUTION

  36. 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

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

  38. 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)

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

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

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

  42. Example Assume a laminar boundary layer has a velocity profile as u(y)=U(y/d) for 0yd 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

  43. 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:

  44. 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.

  45. 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

  46. Momentum Integral Relation for Flat-plate BL Outlet Inlet Continuity

  47. Meantime For flat plate boundary layer

  48. Shape factor

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