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Compressible Frictional Flow Past Wings P M V Subbarao Professor Mechanical Engineering Department I I T Delhi A Small and Significant Region of Curse …….. A continuously Growing Solid affected Region. The Boundary Layer An explicit Negligence by Potential Flow Theory. Great Disadvantage for Simple fluid Systems De Alembert to Prandtl 1822 Ideal to Real 1752 1904 1860 Concept of Solid Fluid Interaction • Perfectly smooth surface (ideal surface) Real surface U2 U1 U1 U2 U2 U U Φ Φ Φ Specular reflection Diffuse reflection • The Momentum & convective heat transfer is defined for a combined solid and fluid system. • The fluid packets close to a solid wall attain a zero relative velocity close to the solid wall : Momentum Boundary Layer. • The fluid packets close to a solid wall come to mechanical equilibrium with the wall. • The fluid particles will exchange maximum possible momentum flux with the solid wall. • A Zero velocity difference exists between wall and fluid packets at the wall. • A small layer of fluid particles close the the wall come to Mechanical, Thermal and Chemical Equilibrium With solid wall. • Fundamentally this fluid layer is in Thermodynamic Equilibrium with the solid wall. Introduction • A boundary layer is a thin region in the fluid adjacent to a surface where velocity, temperature and/or concentration gradients normal to the surface are significant. • Typically, the flow is predominantly in one direction. • As the fluid moves over a surface, a velocity gradient is present in a region known as the velocity boundary layer, δ(x). • Likewise, a temperature gradient forms (T ∞ ≠ Ts) in the thermal boundary layer, δt(x), • Therefore, examine the boundary layer at the surface (y = 0). • Flat Plate Boundary Layer is an hypothetical standard for initiation of basic analysis. Boundary Layer Thickness u y 0.99Ve The governing equations for steady two dimensional incompressible fluid flow with negligible viscous dissipation: Boundary Conditions 0 Ve 0 0 Twall T Define dimensionless variables: x x L * v v u * y y L * p p 2 u * u u u * T Ts T Ts Similarity Parameters: Re u L Pr Pe Re Pr u v * 0 * x y * * 2 * 2 * p 1 u u * u * u u v * 2 * * * *2 x y x Re L x y * * * 2 1 * * u v * * *2 x y Re L Pr y Similarity Solution for Flat Plate Boundary Layer u 1 u * u u v * * *2 x y Re L y * * 2 * * * u * & v * y x 2 2 3 1 * *2 * * * *3 y x y x y Re L y * Similarity variables : f u x * u & u y x * 1 * *2 * * * *3 y x y x y Re L y 2 2 3 Substitute similarity variables: d3 f d2 f 2 3f 0 2 d d This is called as Blasius Equation. An ordinary differential equation with following boundary conditions. df f 0 0 d 0 and df 1 d Numerical Solution of Blasius Equation d3 f d2 f 2 3f 0 2 d d Let f ( ) f1 ( ) df1 ( ) ' ' Let f 2 ( ) f1 f d df 2 ( ) d 2 f1 ( ) '' '' Let f 3 ( ) f1 f 2 d d Substitute in Blasius Equation df 3 2 f1 f 3 0 d df 2 f3 d df1 f2 d Fourth-order Runge-Kutta method Blasius Solution d3 f d2 f 2 3f 0 2 d d 0 0.2 0.4 0.6 f f’ f’’ 0 0.00664 0.02656 0.05974 0 0.06641 0.13277 0.198994 0.3321 0.3320 0.3315 0.8 1.0 2.0 0.10611 0.16557 0.65003 0.26471 0.32979 0.62977 3.0 4.0 5.0 1.39682 2.30576 3.28329 0.84605 0.95552 0.99155 Blasius Similarity Solution • Blasius equation was first solved numerically (undoubtedly by hand 1908). •Conclusions from the Blasius solution: x , and 1 u Variation of Reynolds numbers All Engineering Applications What Sort of Reynolds Numbers do We Encounter in Supersonic Flight? “Transition Line” Space Shuttle Velocity Profile in Boundary Layer V u(y) u V dy y y • Simple Velocity Profile Models Laminar 2y y Ve u( y) u V y Turbulent 2 u( y) y Ve 1 n • Laminar C D fric 4 15 c • Turbulent CD fric 2n c n 1n 2 Turbulent Skin Friction • Turbulent Boundary Layer • No Theoretical Prediction for Boundary Layer Thickness for Turbulent Boundary layer • Statistical Empirical Correlation “Time averaged” 0.16c 1 n R e Compare Laminar and Turbulent Skin Friction CD fric CD fric 32 15 1 Plot these Formulae 1 2 R e 0.32n n 1n 2 1 R e n laminar turbulent Versus Re Compare Laminar and Turbulent Skin Friction Comparison of Reynolds Number and Mach Number V2 2 1 M 2 e 2 Mach number is a measure of the ratio of the fluid Kinetic energy to the fluid internal energy (direct motion To random thermal motion of gas molecules) -- Fundamental Parameter of Compressible Flow 2 Ve 2 c Ve c Ve c 2c Ve 2 Inertial Forces • Re 2 w Ve Viscous Forces Ve Reynold’s number is a measure of the ratio of the Inertial Forces acting on the 2 fluid -- to -- the Viscous Forces Acting on w Ve the fluid -- Fundamental Parameter of Viscous Flow 2 Empirical Skin Friction Correlations “Smooth Flat Plate with No Pressure Gradient” M=0 Re~500,000 Empirical Skin Friction Correlations “Smooth Flat Plate with No Pressure Gradient” 0.32(7) (7) 1(7) 2 1 R e 7 CD fric Plot the laws 1.328 CD fric 1 1 2 R e “exact solution for Laminar Flo” 32 15 1 1 2 R e Simple High Speed Skin Friction Model • So For Our Purposes … we’ll use the “1/7th power” Boundary layer law … and the Exact Laminar Solution Re < 500,000 CD fric 1.328 Exact Blasius Solution CD fric 1 1 2 R e laminar 0.32n n 1 n 2 7 1 1 225 R e 7 R e n turbulent Comparison of Velocity Distributions Supersonic Boundary Layers • When a vehicle travels at Mach numbers greater than one, a significant temperature gradient develops across the boundary layer due to the high levels of viscous dissipation near the wall. • In fact, the static-temperature variation can be very large even in an adiabatic flow, resulting in a low density, high-viscosity region near the wall. • In turn, this leads to a skewed mass-flux profile, a thicker boundary layer, and a region in which viscous effects are somewhat more important than at an equivalent Reynolds number in subsonic flow. • Intuitively, one would expect to see significant dynamical differences between subsonic and supersonic boundary layers. • However, many of these differences can be explained by simply accounting for the fluid-property variations that accompany the temperature variation, as would be the case in a heated incompressible boundary layer. • This suggests a rather passive role for the density differences in these flows, most clearly expressed by Morkovin’s hypothesis. Effect of Mach Number • The friction coefficient is affected by Mach number as well. • This effect is small at subsonic speeds, but becomes appreciable for supersonic aircraft. • The idea is that aerodynamic heating modifies the fluid properties. • For a fully-turbulent flow, the wall temperature may be estimated from: • An effective incompressible temperature ratio is defined: GAS Viscosity Models Sutherland’s Formula Result from kinetic theory that expresses viscosity as a function of temperature T (T ) (Ts ) Ts 3/2 T s Cs T C s 7 C D , firction,turbulent 225 Re1/ 7 CD fric compressible 7 T cV Tavg 225 3/2 T C T s (T ) avg T Tavg C s 7 T cV 225 (T ) Tavg 5/2 Tavg C s T C s C D fric compressible 1 7 CD fric incompressible T Tavg 5 /2 T avg C s T C s 1 7 7 1 7 cV T 225 (T ) Tavg 1 7 5/2 T avg C s T C s 1 7 • What is “Tavg” in the boundary layer? CD fric compressible CD fric incompressible T 5 /2 T C s avg Tavg T Cs 1 7 • Look at small segment of boundary layer, dy • Enthalpy Balance 2 u(y) V u(y) V T T (y) T (y) T 1 2c p 2c p 2c p Ve 2 2 2 V u(y) dy y • Taking the average (Integrating Across Boundary layer) 2 1 2 u(y) V 2 1 V 7 x T 1 T dy 1 dy 2c p 0 Ve 2c p 0 2 Tavg 9 71 x V 2 0 1 T 2c p 9/7 2 V 2 T T 1 9 2c p 2 1 2 M 9 2 V • Valid for Turbulent Flow u(y) dy y Collected Algorithm V c 7 V c C D R e R fric incompressible e 1 7 225 R e CD fric incompressible 0 C D C 120 K for air s 1 fric compressible T 5 /2 T C 7 s avg Tavg T Cs 2 1 2 Tavg T 1 M • Valid for Turbulent Flow 9 2 Skin Friction Versus Mach Number Symmetric Double-wedge Airfoil … L/D (revisited) • Inviscid Analysis t/c • Mach 3 t/c = 0.035 + Symmetric Double-wedgeAirfoil … L/D (revisited) • Blow up of Previous page • Analysis Including skin Friction Model • Mach 3 L/Dmax =7.4 = • Mach 25 • 60 km Altitude L/Dmax =3.18 =